INTERNATIONAL GONGRESS sciimath>\mathrm{K}K-stability involves a strengthened notion called reduced uniform K-stability, which matches the reduced coercivity in (1.14) (see [ 19 , 29 , 39 ] [ 19 , 29 , 39 ] [19,29,39][19,29,39][19,29,39] ). Recall that T ~ T ~ tilde(T)\tilde{\mathbb{T}}T~ denotes a maximal torus of Aut ( X , L ) Aut ⁡ ( X , L ) Aut(X,L)\operatorname{Aut}(X, L)Aut⁡(X,L), and N ~ Q N ~ Q tilde(N)_(Q)\tilde{N}_{\mathbb{Q}}N~Q is defined similar to (1.3).
Definition 2.3. A polarized manifold ( X , L ) ( X , L ) (X,L)(X, L)(X,L) is uniformly K-stable (resp. reduced uniformly K-stable) if there exists γ > 0 γ > 0 gamma > 0\gamma>0γ>0 such that any test configuration ( X , L ) ( X , L ) (X,L)(\mathcal{X}, \mathscr{L})(X,L) satisfies M N A ( X , L ) M N A ( X , L ) ≥ M^(NA)(X,L) >=\mathbf{M}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L}) \geqMNA(X,L)≥ γ J N A ( X , L ) ( γ â‹… J N A ( X , L ) gamma*J^(NA)(X,L)(:}\gamma \cdot \mathbf{J}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L})\left(\right.γ⋅JNA(X,L)( resp. M N A ( X , L ) γ inf ξ N ~ Q J N A ( X ξ , L ξ ) ) M N A ( X , L ) ≥ γ â‹… inf ξ ∈ N ~ Q   J N A X ξ , L ξ {:M^(NA)(X,L) >= gamma*i n f_(xi in tilde(N)_(Q))J^(NA)(X_(xi),L_(xi)))\left.\mathbf{M}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L}) \geq \gamma \cdot \inf _{\xi \in \tilde{N}_{\mathbb{Q}}} \mathbf{J}^{\mathrm{NA}}\left(\mathcal{X}_{\xi}, \mathscr{L}_{\xi}\right)\right)MNA(X,L)≥γ⋅infξ∈N~QJNA(Xξ,Lξ))
Here the twist ( X ξ , L ξ ) X ξ , L ξ (X_(xi),L_(xi))\left(\mathcal{X}_{\xi}, \mathscr{L}_{\xi}\right)(Xξ,Lξ) is introduced by Hisamoto [39]. One way to define it as a test configuration is by resolving the composition of birational morphisms ( X , L ) ( X C = ( X , L ) → X C = (X,L)rarr(X_(C)=:}(\mathcal{X}, \mathscr{L}) \rightarrow\left(X_{\mathbb{C}}=\right.(X,L)→(XC= X × C , L C = p 1 L ) σ ξ ( X C , L C ) X × C , L C = p 1 ∗ L → σ ξ X C , L C {:X xxC,L_(C)=p_(1)^(**)L)rarr"sigma_(xi)"(X_(C),L_(C))\left.X \times \mathbb{C}, L_{\mathbb{C}}=p_{1}^{*} L\right) \xrightarrow{\sigma_{\xi}}\left(X_{\mathbb{C}}, L_{\mathbb{C}}\right)X×C,LC=p1∗L)→σξ(XC,LC) where σ ξ σ ξ sigma_(xi)\sigma_{\xi}σξ is the C C ∗ C^(**)\mathbb{C}^{*}C∗-action generated by ξ ξ xi\xiξ. Alternatively, it can be defined in a more general setting of filtrations (see Example 2.8).

2.2. Non-Archimedean pluripotential theory

We discuss how non-Archimedean pluripotential theory as developed by Boucksom-Jonsson can be applied to study K-stability. Corresponding to a regularization result in the complex analytic case, an u.s.c. function ϕ : X N A R { + } Ï• : X N A → R ∪ { + ∞ } phi:X^(NA)rarrRuu{+oo}\phi: X^{\mathrm{NA}} \rightarrow \mathbb{R} \cup\{+\infty\}Ï•:XNA→R∪{+∞} is called a nonArchimedean psh potential if it is a decreasing limit of a sequence from H N A H N A H^(NA)\mathscr{H}^{\mathrm{NA}}HNA. Denote the space of such functions by P S H N A P S H N A PSH^(NA)\mathrm{PSH}^{\mathrm{NA}}PSHNA. Boucksom-Jonsson introduced the following nonArchimedean version of the finite energy space. First corresponding to (1.10), for any ϕ Ï• ∈ phi in\phi \inϕ∈ P S H N A P S H N A PSH^(NA)\mathrm{PSH}^{\mathrm{NA}}PSHNA, define
E N A ( ϕ ) = inf { E N A ( ϕ ~ ) : ϕ ~ ϕ , ϕ ~ H N A } E N A ( Ï• ) = inf E N A ( Ï• ~ ) : Ï• ~ ≥ Ï• , Ï• ~ ∈ H N A E^(NA)(phi)=i n f{E^(NA)(( tilde(phi))):( tilde(phi)) >= phi,( tilde(phi))inH^(NA)}\mathbf{E}^{\mathrm{NA}}(\phi)=\inf \left\{\mathbf{E}^{\mathrm{NA}}(\tilde{\phi}): \tilde{\phi} \geq \phi, \tilde{\phi} \in \mathscr{H}^{\mathrm{NA}}\right\}ENA(Ï•)=inf{ENA(Ï•~):Ï•~≥ϕ,Ï•~∈HNA}
Then, corresponding to (1.11), define the space of non-Archimedean finite energy potentials by
( ε 1 ) N A = { ϕ P S H N A : E N A ( ϕ ) > } ε 1 N A = Ï• ∈ P S H N A : E N A ( Ï• ) > − ∞ (epsi^(1))^(NA)={phi inPSH^(NA):E^(NA)(phi) > -oo}\left(\varepsilon^{1}\right)^{\mathrm{NA}}=\left\{\phi \in \mathrm{PSH}^{\mathrm{NA}}: \mathbf{E}^{\mathrm{NA}}(\phi)>-\infty\right\}(ε1)NA={ϕ∈PSHNA:ENA(Ï•)>−∞}
This space is again equipped with a strong topology which makes E N A E N A E^(NA)\mathbf{E}^{\mathrm{NA}}ENA continuous. Boucksom-Jonsson showed in [22] that the non-Archimedean Monge-Ampère measure M A N A ( ϕ ) M A N A ( Ï• ) MA^(NA)(phi)\mathrm{MA}^{\mathrm{NA}}(\phi)MANA(Ï•) is well defined for any ϕ ( E 1 ) N A Ï• ∈ E 1 N A phi in(E^(1))^(NA)\phi \in\left(\mathcal{E}^{1}\right)^{\mathrm{NA}}ϕ∈(E1)NA such that if { ϕ k } k N H N A Ï• k k ∈ N ⊂ H N A {phi_(k)}_(k inN)subH^(NA)\left\{\phi_{k}\right\}_{k \in \mathbb{N}} \subset \mathscr{H}^{\mathrm{NA}}{Ï•k}k∈N⊂HNA converges to ϕ Ï• phi\phiÏ• strongly, then M A N A ( ϕ k ) M A N A Ï• k MA^(NA)(phi_(k))\mathrm{MA}^{\mathrm{NA}}\left(\phi_{k}\right)MANA(Ï•k) converges to M A N A ( ϕ ) M A N A ( Ï• ) MA^(NA)(phi)\mathrm{MA}^{\mathrm{NA}}(\phi)MANA(Ï•) weakly.
A large class of potentials come from filtrations (see [19]). Set R m = H 0 ( X , m L ) R m = H 0 ( X , m L ) R_(m)=H^(0)(X,mL)R_{m}=H^{0}(X, m L)Rm=H0(X,mL).
Definition 2.4. A filtration is the data F = { F λ R m R m ; λ R , m N } F = F λ R m ⊆ R m ; λ ∈ R , m ∈ N F={F^(lambda)R_(m)subeR_(m);lambda inR,m inN}\mathcal{F}=\left\{\mathcal{F}^{\lambda} R_{m} \subseteq R_{m} ; \lambda \in \mathbb{R}, m \in \mathbb{N}\right\}F={FλRm⊆Rm;λ∈R,m∈N} that satisfies the following four conditions:
(i) F λ R m F λ R m F λ R m ⊆ F λ ′ R m F^(lambda)R_(m)subeF^(lambda^('))R_(m)\mathcal{F}^{\lambda} R_{m} \subseteq \mathcal{F}^{\lambda^{\prime}} R_{m}FλRm⊆Fλ′Rm, if λ λ λ ≥ λ ′ lambda >= lambda^(')\lambda \geq \lambda^{\prime}λ≥λ′; (ii) F λ R m = λ < λ F λ R m F λ R m = â‹‚ λ ′ < λ   F λ ′ R m F^(lambda)R_(m)=nnn_(lambda^(') < lambda)F^(lambda^('))R_(m)\mathcal{F}^{\lambda} R_{m}=\bigcap_{\lambda^{\prime}<\lambda} \mathcal{F}^{\lambda^{\prime}} R_{m}FλRm=⋂λ′<λFλ′Rm;
(iii) F λ R m F λ R m F λ + λ R m + m F λ R m â‹… F λ ′ R m ′ ⊆ F λ + λ ′ R m + m ′ F^(lambda)R_(m)*F^(lambda^('))R_(m^('))subeF^(lambda+lambda^('))R_(m+m^('))\mathscr{F}^{\lambda} R_{m} \cdot \mathcal{F}^{\lambda^{\prime}} R_{m^{\prime}} \subseteq \mathcal{F}^{\lambda+\lambda^{\prime}} R_{m+m^{\prime}}FλRmâ‹…Fλ′Rm′⊆Fλ+λ′Rm+m′, for λ , λ R λ , λ ′ ∈ R lambda,lambda^(')inR\lambda, \lambda^{\prime} \in \mathbb{R}λ,λ′∈R and m , m N m , m ′ ∈ N m,m^(')inNm, m^{\prime} \in \mathbb{N}m,m′∈N;
(iv) There exist e , e + Z e − , e + ∈ Z e_(-),e_(+)inZe_{-}, e_{+} \in \mathbb{Z}e−,e+∈Z such that F m e R m = R m F m e − R m = R m F^(me)_(-)R_(m)=R_(m)\mathcal{F}^{m e}{ }_{-} R_{m}=R_{m}Fme−Rm=Rm and F m e + R m = 0 F m e + R m = 0 F^(me_(+))R_(m)=0\mathcal{F}^{m e_{+}} R_{m}=0Fme+Rm=0 for m m ∈ m inm \inm∈ Z 0 Z ≥ 0 Z_( >= 0)\mathbb{Z}_{\geq 0}Z≥0.
Filtration F F F\mathscr{F}F is finitely generated if its extended Rees algebra R ( F ) R ( F ) R(F)\mathscr{R}(\mathscr{F})R(F) is finitely generated where
R ( F ) = λ R m N t λ F λ R m R ( F ) = ⨁ λ ∈ R   ⨁ m ∈ N   t − λ F λ R m R(F)=bigoplus_(lambda inR)bigoplus_(m inN)t^(-lambda)F^(lambda)R_(m)\mathcal{R}(\mathcal{F})=\bigoplus_{\lambda \in \mathbb{R}} \bigoplus_{m \in \mathbb{N}} t^{-\lambda} \mathscr{F}^{\lambda} R_{m}R(F)=⨁λ∈R⨁m∈Nt−λFλRm
In this case F F F\mathcal{F}F induces a degeneration of X X XXX into X 0 = Proj ( m , λ F λ R m / F > λ R m ) X 0 = Proj ⁡ ⨁ m , λ   F λ R m / F > λ R m X_(0)=Proj(bigoplus_(m,lambda)F^(lambda)R_(m)//F > lambdaR_(m))\mathcal{X}_{0}=\operatorname{Proj}\left(\bigoplus_{m, \lambda} \mathcal{F}^{\lambda} R_{m} / \mathcal{F}>\lambda R_{m}\right)X0=Proj⁡(⨁m,λFλRm/F>λRm).
For a general F , { F λ R ; λ R } F , F λ R â„“ ; λ ∈ R F,{FlambdaR_(â„“);lambda inR}\mathscr{F},\left\{\mathcal{F} \lambda R_{\ell} ; \lambda \in \mathbb{R}\right\}F,{FλRâ„“;λ∈R} generates a filtration F ( ) F ( â„“ ) F(â„“)\mathscr{\mathscr { F }}(\ell)F(â„“) on R ( ) := m N R m R ( â„“ ) := ⨁ m ∈ N   R m â„“ R^((â„“)):=bigoplus_(m inN)R_(mâ„“)R^{(\ell)}:=\bigoplus_{m \in \mathbb{N}} R_{m \ell}R(â„“):=⨁m∈NRmâ„“, which induces a non-Archimedean psh potential ϕ ˇ ( ) H N A Ï• ˇ ( â„“ ) ∈ H N A phi^(ˇ)^((â„“))inH^(NA)\check{\phi}^{(\ell)} \in \mathscr{H}^{\mathrm{NA}}ϕˇ(â„“)∈HNA. Define
ϕ F = ( lim sup + ϕ ˇ ( ) ) Ï• F = lim sup â„“ → + ∞   Ï• ˇ ( â„“ ) ∗ phi_(F)=(l i m   s u p_(â„“rarr+oo)phi^(ˇ)^((â„“)))^(**)\phi_{\mathcal{F}}=\left(\limsup _{\ell \rightarrow+\infty} \check{\phi}^{(\ell)}\right)^{*}Ï•F=(lim supℓ→+∞ϕˇ(â„“))∗
where ( ) ( ⋅ ) ∗ (*)^(**)(\cdot)^{*}(⋅)∗ denotes the upper-semicontinuous regularization.
Example 2.5. Filtration F F F\mathscr{F}F is a Z Z Z\mathbb{Z}Z-filtration if F λ R m = F λ R m F λ R m = F ⌈ λ ⌉ R m F^(lambda)R_(m)=F|~lambda~|R_(m)\mathscr{F}^{\lambda} R_{m}=\mathscr{F}\lceil\lambda\rceil R_{m}FλRm=F⌈λ⌉Rm. By [ 19 , 63 , 69 ] [ 19 , 63 , 69 ] [19,63,69][19,63,69][19,63,69], there is a one-to-one correspondence between test configurations equipped with relatively ample Q Q Q\mathbb{Q}Q polarizations and finitely generated Z Z Z\mathbb{Z}Z-filtrations. Any test configuration ( X , L ) ( X , L ) (X,L)(\mathcal{X}, \mathscr{L})(X,L) defines such a filtration by
Conversely, if F F F\mathscr{F}F is a finitely generated Z Z Z\mathbb{Z}Z-filtration, then ( X := Proj C [ t ] ( R ( F ˇ ( ) ) ) , 1 O X ( 1 ) ) X := Proj C [ t ] ⁡ ( R ( F ˇ ( â„“ ) ) ) , 1 â„“ O X ( 1 ) (X:=Proj_(C[t])(R((F^(ˇ))(â„“))),(1)/(â„“)O_(X)(1))\left(\mathcal{X}:=\operatorname{Proj}_{\mathbb{C}[t]}(\mathcal{R}(\check{\mathcal{F}}(\ell))), \frac{1}{\ell} \mathcal{O}_{X}(1)\right)(X:=ProjC[t]⁡(R(Fˇ(â„“))),1â„“OX(1)) is a test configuration for â„“ â„“\ellâ„“ sufficiently divisible.
Example 2.6. In Definition 2.1 of test configurations, if we do not require L L L\mathscr{L}L to be π Ï€ pi\piÏ€ semiample, then we call ( X , L ) ( X , L ) (X,L)(\mathcal{X}, \mathscr{L})(X,L) a model (of ( X × C , p 1 L ) X × C , p 1 ∗ L (X xxC,p_(1)^(**)L)\left(X \times \mathbb{C}, p_{1}^{*} L\right)(X×C,p1∗L) ). The same definition in (2.7) defines a filtration also denoted by F ( x , L ) F ( x , L ) F_((x,L))\mathscr{F}_{(x, \mathscr{L})}F(x,L). However, in general the filtration is not finitely generated anymore. Fix any model ( X , L ) ( X , L ) (X,L)(\mathcal{X}, \mathscr{L})(X,L) such that L ¯ L ¯ bar(L)\overline{\mathscr{L}}L¯ is big over X ¯ X ¯ bar(X)\bar{X}X¯ (we call such ( X , L ) ( X , L ) (X,L)(\mathcal{X}, \mathscr{L})(X,L) a big model for ( X , L ) ( X , L ) (X,L)(X, L)(X,L) ). In [46] we obtained the following formula for the non-Archimedean Monge-Ampère measure of ϕ = ϕ ( X , L ) := ϕ F ( X , L ) Ï• = Ï• ( X , L ) := Ï• F ( X , L ) phi=phi_((X,L)):=phi_(F_((X,L)))\phi=\phi_{(X, \mathscr{L})}:=\phi_{\mathcal{F}_{(X, \mathscr{L})}}Ï•=Ï•(X,L):=Ï•F(X,L) which generalizes (2.5):
(2.8) MA N A ( ϕ ) = i b i ( L n F i ) δ v i (2.8) MA N A ⁡ ( Ï• ) = ∑ i   b i L â‹… n â‹… F i δ v i {:(2.8)MA^(NA)(phi)=sum_(i)b_(i)((:L^(*n):)*F_(i))delta_(v_(i)):}\begin{equation*} \operatorname{MA}^{\mathrm{NA}}(\phi)=\sum_{i} b_{i}\left(\left\langle\mathscr{L}^{\cdot n}\right\rangle \cdot F_{i}\right) \delta_{v_{i}} \tag{2.8} \end{equation*}(2.8)MANA⁡(Ï•)=∑ibi(⟨Lâ‹…n⟩⋅Fi)δvi
Here for any divisor D D DDD, we use the notion of a positive intersection product introduced in [17]:
L ¯ n + 1 := vol ( L ¯ ) = lim m + h 0 ( X ¯ , m L ¯ ) m n + 1 ( n + 1 ) ! , L ¯ n D := 1 n + 1 d d t | t = 0 vol ( L ¯ + t D ) L ¯ n + 1 := vol ⁡ ( L ¯ ) = lim m → + ∞   h 0 ( X ¯ , m L ¯ ) m n + 1 ( n + 1 ) ! , L ¯ n â‹… D := 1 n + 1 d d t t = 0 vol ⁡ ( L ¯ + t D ) (: bar(L)^(n+1):):=vol( bar(L))=lim_(m rarr+oo)(h^(0)(( bar(X)),m bar(L)))/((m^(n+1))/((n+1)!)),quad(: bar(L)^(n):)*D:=(1)/(n+1)(d)/(dt)|_(t=0)vol( bar(L)+tD)\left\langle\bar{L}^{n+1}\right\rangle:=\operatorname{vol}(\overline{\mathscr{L}})=\lim _{m \rightarrow+\infty} \frac{h^{0}(\bar{X}, m \overline{\mathscr{L}})}{\frac{m^{n+1}}{(n+1)!}}, \quad\left\langle\bar{L}^{n}\right\rangle \cdot D:=\left.\frac{1}{n+1} \frac{d}{d t}\right|_{t=0} \operatorname{vol}(\overline{\mathscr{L}}+t D)⟨L¯n+1⟩:=vol⁡(L¯)=limm→+∞h0(X¯,mL¯)mn+1(n+1)!,⟨L¯n⟩⋅D:=1n+1ddt|t=0vol⁡(L¯+tD)
Example 2.7. Any v X Q div v ∈ X Q div  v inX_(Q)^("div ")v \in X_{\mathbb{Q}}^{\text {div }}v∈XQdiv  defines a filtration: for any λ R λ ∈ R lambda inR\lambda \in \mathbb{R}λ∈R and m Z 0 m ∈ Z ≥ 0 m inZ_( >= 0)m \in \mathbb{Z}_{\geq 0}m∈Z≥0, define
(2.9) F v λ R m = { s R m : v ( s ) λ } (2.9) F v λ R m = s ∈ R m : v ( s ) ≥ λ {:(2.9)F_(v)^(lambda)R_(m)={s inR_(m):v(s) >= lambda}:}\begin{equation*} \mathscr{F}_{v}^{\lambda} R_{m}=\left\{s \in R_{m}: v(s) \geq \lambda\right\} \tag{2.9} \end{equation*}(2.9)FvλRm={s∈Rm:v(s)≥λ}
Boucksom-Jonsson proved in [21] that M A N A ( ϕ F v ) = V δ v M A N A Ï• F v = V â‹… δ v MA^(NA)(phi_(F_(v)))=V*delta_(v)\mathrm{MA}^{\mathrm{NA}}\left(\phi_{\mathcal{F}_{v}}\right)=\mathbf{V} \cdot \delta_{v}MANA(Ï•Fv)=V⋅δv.
Example 2.8. Assume a torus T ~ ( C ) r T ~ ≅ C ∗ r tilde(T)~=(C^(**))^(r)\tilde{\mathbb{T}} \cong\left(\mathbb{C}^{*}\right)^{r}T~≅(C∗)r-acts on ( X , L ) ( X , L ) (X,L)(X, L)(X,L). Then we have a weight decomposition R m = α Z r R m , α R m = ⨁ α ∈ Z r   R m , α R_(m)=bigoplus_(alpha inZ^(r))R_(m,alpha)R_{m}=\bigoplus_{\alpha \in \mathbb{Z}^{r}} R_{m, \alpha}Rm=⨁α∈ZrRm,α. For any ξ N ~ R ξ ∈ N ~ R xi in tilde(N)_(R)\xi \in \tilde{N}_{\mathbb{R}}ξ∈N~R, we can define the ξ ξ xi\xiξ-twist of a given filtration: F ξ λ R m = F λ α , ξ R m , α F ξ λ R m = F λ − ⟨ α , ξ ⟩ R m , α F_(xi)^(lambda)R_(m)=F^(lambda-(:alpha,xi:))R_(m,alpha)\mathcal{F}_{\xi}^{\lambda} R_{m}=\mathcal{F}^{\lambda-\langle\alpha, \xi\rangle} R_{m, \alpha}FξλRm=Fλ−⟨α,ξ⟩Rm,α. On the other hand, there is an induced N ~ R N ~ R tilde(N)_(R)\tilde{N}_{\mathbb{R}}N~R-action on ( X N A ) T ~ X N A T ~ (X^(NA))^( tilde(T))\left(X^{\mathrm{NA}}\right)^{\tilde{\mathbb{T}}}(XNA)T~ which sends ( ξ , v ) ( ξ , v ) (xi,v)(\xi, v)(ξ,v) to v ξ ( X N A ) T ~ v ξ ∈ X N A T ~ v_(xi)in(X^(NA))^( tilde(T))v_{\xi} \in\left(X^{\mathrm{NA}}\right)^{\tilde{\mathbb{T}}}vξ∈(XNA)T~ determined by the following condition: if f C ( X ) α f ∈ C ( X ) α f inC(X)_(alpha)f \in \mathbb{C}(X)_{\alpha}f∈C(X)α which means f t 1 = t α f f ∘ t − 1 = t α â‹… f f@t^(-1)=t^(alpha)*ff \circ \mathrm{t}^{-1}=\mathrm{t}^{\alpha} \cdot ff∘t−1=tα⋅f for any t T ~ t ∈ T ~ tin tilde(T)\mathrm{t} \in \tilde{\mathbb{T}}t∈T~, then v ξ ( f ) = α , ξ + v ( f ) v ξ ( f ) = ⟨ α , ξ ⟩ + v ( f ) v_(xi)(f)=(:alpha,xi:)+v(f)v_{\xi}(f)=\langle\alpha, \xi\rangle+v(f)vξ(f)=⟨α,ξ⟩+v(f). We then have the following formula: M A N A ( ϕ F ξ ) = ( ξ ) M A N A ( ϕ F ) M A N A Ï• F ξ = ( − ξ ) ∗ M A N A Ï• F MA^(NA)(phi_(F_(xi)))=(-xi)_(**)MA^(NA)(phi_(F))\mathrm{MA}^{\mathrm{NA}}\left(\phi_{\mathscr{F}_{\xi}}\right)=(-\xi)_{*} \mathrm{MA}^{\mathrm{NA}}\left(\phi_{\mathcal{F}}\right)MANA(Ï•Fξ)=(−ξ)∗MANA(Ï•F) (see [ 44 , 45 ] [ 44 , 45 ] [44,45][44,45][44,45] ).
Generalizing the case of test configurations, Boucksom-Jonsson showed that the non-Archimedean functionals from (2.2)-(2.4) are well defined for all ϕ ( E 1 ) N A Ï• ∈ E 1 N A phi in(E^(1))^(NA)\phi \in\left(\mathcal{E}^{1}\right)^{\mathrm{NA}}ϕ∈(E1)NA by using integrals over X N A X N A X^(NA)X^{\mathrm{NA}}XNA mentioned before (for example, for H N A H N A H^(NA)\mathbf{H}^{\mathrm{NA}}HNA use (2.6)).
Example 2.9. For any filtration F F F\mathcal{F}F, it is known that ϕ F ( E 1 ) N A Ï• F ∈ E 1 N A phi_(F)in(E^(1))^(NA)\phi_{\mathcal{F}} \in\left(\mathcal{E}^{1}\right)^{\mathrm{NA}}Ï•F∈(E1)NA. Following [19], define
vol ( F ( t ) ) = lim m + dim C F m t R m m n / n ! vol ⁡ F ( t ) = lim m → + ∞   dim C ⁡ F m t R m m n / n ! vol(F^((t)))=lim_(m rarr+oo)(dim_(C)F^(mt)R_(m))/(m^(n)//n!)\operatorname{vol}\left(\mathcal{F}^{(t)}\right)=\lim _{m \rightarrow+\infty} \frac{\operatorname{dim}_{\mathbb{C}} \mathcal{F}^{m t} R_{m}}{m^{n} / n!}vol⁡(F(t))=limm→+∞dimC⁡FmtRmmn/n!
Then E N A E N A E^(NA)\mathbf{E}^{\mathrm{NA}}ENA is the following "expected vanishing order" with respect to F F F\mathscr{F}F (see [21]).
(2.10) E N A ( ϕ F ) = 1 V R t ( d vol ( F ( t ) ) ) (2.10) E N A Ï• F = 1 V ∫ R   t − d vol ⁡ F ( t ) {:(2.10)E^(NA)(phi_(F))=(1)/(V)int_(R)t(-d vol(F^((t)))):}\begin{equation*} \mathbf{E}^{\mathrm{NA}}\left(\phi_{\mathcal{F}}\right)=\frac{1}{\mathbf{V}} \int_{\mathbb{R}} t\left(-d \operatorname{vol}\left(\mathscr{F}^{(t)}\right)\right) \tag{2.10} \end{equation*}(2.10)ENA(Ï•F)=1V∫Rt(−dvol⁡(F(t)))
Similar to Theorem 1.2, we also have important regularization properties:
Theorem 2.10 ([22]). For any ϕ ( E 1 ) N A Ï• ∈ E 1 N A phi in(E^(1))^(NA)\phi \in\left(\mathcal{E}^{1}\right)^{\mathrm{NA}}ϕ∈(E1)NA, there exists { ϕ k } k N H N A Ï• k k ∈ N ⊂ H N A {phi_(k)}_(k inN)subH^(NA)\left\{\phi_{k}\right\}_{k \in \mathbb{N}} \subset \mathscr{H}^{\mathrm{NA}}{Ï•k}k∈N⊂HNA (i.e., ϕ k = ϕ ( X k , L k ) Ï• k = Ï• X k , L k phi_(k)=phi_((X_(k),L_(k)))\phi_{k}=\phi_{\left(X_{k}, \mathscr{L}_{k}\right)}Ï•k=Ï•(Xk,Lk) for a test configuration ( X k , L k ) ) X k , L k {:(X_(k),L_(k)))\left.\left(\mathcal{X}_{k}, \mathscr{L}_{k}\right)\right)(Xk,Lk)) such that ϕ k ϕ Ï• k → Ï• phi_(k)rarr phi\phi_{k} \rightarrow \phiÏ•k→ϕ in the strong topology and F N A ( ϕ k ) F N A Ï• k → F^(NA)(phi_(k))rarr\mathbf{F}^{\mathrm{NA}}\left(\phi_{k}\right) \rightarrowFNA(Ï•k)→ F N A ( ϕ ) F N A ( Ï• ) F^(NA)(phi)\mathbf{F}^{\mathrm{NA}}(\phi)FNA(Ï•) for F { E , Λ , E K X } F ∈ E , Λ , E K X Fin{E,Lambda,E^(K_(X))}\mathbf{F} \in\left\{\mathbf{E}, \boldsymbol{\Lambda}, \mathbf{E}^{K_{X}}\right\}F∈{E,Λ,EKX}.
Boucksom-Jonsson conjectured that the same conclusion should also hold for H N A H N A H^(NA)\mathbf{H}^{\mathrm{NA}}HNA. This conjecture is still open in general and it is important in the non-Archimedean approach to the YTD conjecture. We have made progress in this direction.
Theorem 2.11 ([45,46]). (1) For any ϕ ( E 1 ) N A Ï• ∈ E 1 N A phi in(E^(1))^(NA)\phi \in\left(\mathcal{E}^{1}\right)^{\mathrm{NA}}ϕ∈(E1)NA, there exist models { ( X k , L k ) } k N X k , L k k ∈ N {(X_(k),L_(k))}_(k inN)\left\{\left(\mathcal{X}_{k}, \mathscr{L}_{k}\right)\right\}_{k \in \mathbb{N}}{(Xk,Lk)}k∈N such that ϕ k = ϕ ( x k , L k ) ϕ Ï• k = Ï• x k , L k → Ï• phi_(k)=phi_((x_(k),L_(k)))rarr phi\phi_{k}=\phi_{\left(x_{k}, \mathscr{L}_{k}\right)} \rightarrow \phiÏ•k=Ï•(xk,Lk)→ϕ in the strong topology and H N A ( ϕ k ) H N A ( ϕ ) H N A Ï• k → H N A ( Ï• ) H^(NA)(phi_(k))rarrH^(NA)(phi)\mathbf{H}^{\mathrm{NA}}\left(\phi_{k}\right) \rightarrow \mathbf{H}^{\mathrm{NA}}(\phi)HNA(Ï•k)→HNA(Ï•).
(2) For any big model ( X , L ) ( X , L ) (X,L)(\mathcal{X}, \mathscr{L})(X,L), we have the following formula that generalizes (2.4):
M N A ( X , L ) = 1 V L n K x ¯ / P 1 + S _ ( n + 1 ) V L n + 1 M N A ( X , L ) = 1 V L â‹… n â‹… K x ¯ / P 1 + S _ ( n + 1 ) V L â‹… n + 1 M^(NA)(X,L)=(1)/(V)(:L^(*n):)*K_( bar(x)//P^(1))+(S_)/((n+1)V)(:L^(*n+1):)\mathbf{M}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L})=\frac{1}{\mathbf{V}}\left\langle\mathscr{L}^{\cdot n}\right\rangle \cdot K_{\bar{x} / \mathbb{P}^{1}}+\frac{\underline{S}}{(n+1) \mathbf{V}}\left\langle\mathscr{L}^{\cdot n+1}\right\rangleMNA(X,L)=1V⟨Lâ‹…n⟩⋅Kx¯/P1+S_(n+1)V⟨Lâ‹…n+1⟩
The idea for proving the first statement is similar to the Archimedean setting in [8]. First we regularize the measure M A N A ( ϕ ) M A N A ( Ï• ) MA^(NA)(phi)\mathrm{MA}^{\mathrm{NA}}(\phi)MANA(Ï•) with converging entropy. In fact, we find a way to regularize it by using measures supported at finitely many points in X Q div X Q div  X_(Q)^("div ")X_{\mathbb{Q}}^{\text {div }}XQdiv . Then we use the solution of non-Archimedean Monge-Ampère equations obtained in [18] to get the wanted potentials which are known to be associated to models. However, in the non-Archimedean case, there is not yet a characterization of measures associated to test configurations which prevents us from regularizing via test configurations. The second statement in Theorem 2.11 follows from the formula (2.8), and it prompts us to propose the following algebro-geometric conjecture which would strengthen the classical Fujita approximation theorem.
Conjecture 2.12. Let X ¯ X ¯ bar(X)\overline{\mathcal{X}}X¯ be a smooth ( n + 1 ) ( n + 1 ) (n+1)(n+1)(n+1)-dimensional projective variety. Let L ¯ L ¯ bar(L)\bar{L}L¯ be a big line bundle over X ¯ X ¯ bar(X)\bar{X}X¯. Then there exist birational morphisms μ k : X ¯ k X ¯ μ k : X ¯ k → X ¯ mu_(k): bar(X)_(k)rarr bar(X)\mu_{k}: \bar{X}_{k} \rightarrow \bar{X}μk:X¯k→X¯ and decompositions μ k L ¯ = L ¯ k + E k μ k ∗ L ¯ = L ¯ k + E k mu_(k)^(**) bar(L)= bar(L)_(k)+E_(k)\mu_{k}^{*} \overline{\mathscr{L}}=\overline{\mathscr{L}}_{k}+E_{k}μk∗L¯=L¯k+Ek in N 1 ( X ¯ ) Q N 1 ( X ¯ ) Q N^(1)( bar(X))_(Q)N^{1}(\overline{\mathcal{X}})_{\mathbb{Q}}N1(X¯)Q with L ¯ k L ¯ k bar(L)_(k)\overline{\mathcal{L}}_{k}L¯k semiample and E k E k E_(k)E_{k}Ek effective such that
lim k + L ¯ k n + 1 = vol ( L ¯ ) , lim k + L ¯ k n K x ¯ k = 1 n + 1 d d t vol ( L ¯ + t K x ¯ ) | t = 0 = L ¯ n K x ¯ lim k → + ∞   L ¯ k n + 1 = vol ⁡ ( L ¯ ) , lim k → + ∞   L ¯ k n â‹… K x ¯ k = 1 n + 1 d d t vol ⁡ ( L ¯ + t K x ¯ ) t = 0 = L ¯ â‹… n â‹… K x ¯ lim_(k rarr+oo) bar(L)_(k)^(n+1)=vol( bar(L)),quadlim_(k rarr+oo) bar(L)_(k)^(n)*K bar(x)_(k)=(1)/(n+1)(d)/(dt)vol( bar(L)+tK( bar(x)))|_(t=0)=(: bar(L)^(*n):)*K bar(x)\lim _{k \rightarrow+\infty} \overline{\mathscr{L}}_{k}^{n+1}=\operatorname{vol}(\overline{\mathscr{L}}), \quad \lim _{k \rightarrow+\infty} \overline{\mathscr{L}}_{k}^{n} \cdot K \bar{x}_{k}=\left.\frac{1}{n+1} \frac{d}{d t} \operatorname{vol}(\overline{\mathscr{L}}+t K \bar{x})\right|_{t=0}=\left\langle\bar{L}^{\cdot n}\right\rangle \cdot K \bar{x}limk→+∞L¯kn+1=vol⁡(L¯),limk→+∞L¯knâ‹…Kx¯k=1n+1ddtvol⁡(L¯+tKx¯)|t=0=⟨L¯⋅n⟩⋅Kx¯.
It is easy to show that this conjecture is true if L ¯ L ¯ bar(L)\overline{\mathscr{L}}L¯ admits a birational Zariski decomposition. The author verified this conjecture for certain examples of big line bundles due to Nakamaya which do not admit such decompositions (see [46]). Y. Odaka observed that when ( X , L ) ( X , L ) (X,L)(\mathcal{X}, \mathscr{L})(X,L) is a big model for a polarized spherical manifold (for example, a polarized toric manifold), X ¯ X ¯ bar(X)\bar{X}X¯ is a Mori dream space which implies that L ¯ L ¯ bar(L)\overline{\mathscr{L}}L¯ admits a Zariski decomposition and hence the above conjecture holds true.

2.3. Stability of Fano varieties

In this section, we assume that X X XXX is a Q Q Q\mathbb{Q}Q-Fano variety (i.e., K X − K X -K_(X)-K_{X}−KX is an ample Q Q Q\mathbb{Q}Q line bundle and X X XXX has at worst klt singularities). Corresponding to (1.9), we have a nonArchimedean D functional. For general test configurations, it first appeared in Berman's work [4] and was reformulated in [19] using non-Archimedean potentials:
L N A ( X , L ) = inf v X Q d i v ( A X ( v ) + ϕ ( X , L ) ( v ) ) , D N A ( X , L ) = E N A ( X , L ) + L N A ( X , L ) L N A ( X , L ) = inf v ∈ X Q d i v   A X ( v ) + Ï• ( X , L ) ( v ) , D N A ( X , L ) = − E N A ( X , L ) + L N A ( X , L ) L^(NA)(X,L)=i n f_(v inX_(Q)^(div))(A_(X)(v)+phi_((X,L))(v)),quadD^(NA)(X,L)=-E^(NA)(X,L)+L^(NA)(X,L)\mathbf{L}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L})=\inf _{v \in X_{\mathbb{Q}}^{\mathrm{div}}}\left(A_{X}(v)+\phi_{(X, \mathscr{L})}(v)\right), \quad \mathbf{D}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L})=-\mathbf{E}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L})+\mathbf{L}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L})LNA(X,L)=infv∈XQdiv(AX(v)+Ï•(X,L)(v)),DNA(X,L)=−ENA(X,L)+LNA(X,L)
The notions of Ding-stability and uniform Ding-stability are defined if M N A M N A M^(NA)\mathbf{M}^{\mathrm{NA}}MNA is replaced by D N A D N A D^(NA)\mathbf{D}^{\mathrm{NA}}DNA in Definitions 2.2 and 2.3. In general, we have the inequality M N A ( X , L ) D N A ( X , L ) M N A ( X , L ) ≥ D N A ( X , L ) M^(NA)(X,L) >= D^(NA)(X,L)\mathbf{M}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L}) \geq \mathbf{D}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L})MNA(X,L)≥DNA(X,L). For Fano varieties, special test configurations play important roles. A test configuration ( X , L ) ( X , L ) (X,L)(\mathcal{X}, \mathscr{L})(X,L) is called special if the central fiber X 0 X 0 X_(0)\mathcal{X}_{0}X0 is a Q Q Q\mathbb{Q}Q-Fano variety and L = K X / P 1 L = − K X / P 1 L=-K_(X//P^(1))\mathscr{L}=-K_{X / \mathbb{P}^{1}}L=−KX/P1. For special test configurations, we have D N A = M N A = E N A = D N A = M N A = − E N A = D^(NA)=M^(NA)=-E^(NA)=\mathbf{D}^{\mathrm{NA}}=\mathbf{M}^{\mathrm{NA}}=-\mathbf{E}^{\mathrm{NA}}=DNA=MNA=−ENA= : Fut x 0 ( ξ ) x 0 ( ξ ) x_(0)(xi)x_{0}(\xi)x0(ξ), the last quantity being the Futaki invariant on X 0 X 0 X_(0)\mathcal{X}_{0}X0 for the holomorphic vector field ξ ξ xi\xiξ that generates the C C ∗ C^(**)\mathbb{C}^{*}C∗-action. The importance of special test configurations was first pointed out in Tian's work [64] motivated by compactness results from metric geometry. The following results show their importance from the point of view of algebraic geometry:
Theorem 2.13 ([35,44,52], see also [ 7 , 21 ] ) [ 7 , 21 ] ) [7,21])[7,21])[7,21]). For any Q Q Q\mathbb{Q}Q-Fano variety, K K KKK-stability is equivalent to Ding-stability, and they are equivalent to K K KKK-stability or Ding-stability over special test configurations. Moreover, the same conclusion holds true if stability is replaced by semistability, polystability, or reduced uniform stability.
The proofs of these results depend on a careful process of Minimal Model Program first used in [52] to transform any given test configuration into a special one. Moreover, crucial calculations show that the relevant invariants such as M N A M N A M^(NA)\mathbf{M}^{\mathrm{NA}}MNA or D N A D N A D^(NA)\mathbf{D}^{\mathrm{NA}}DNA decrease along the MMP process. Theorem 2.13 leads directly to a valuative criterion for K-stability. To state it, first define for any v X Q div v ∈ X Q div  v inX_(Q)^("div ")v \in X_{\mathbb{Q}}^{\text {div }}v∈XQdiv  an invariant (see Example 2.9):
(2.11) S L ( v ) := 1 V 0 + vol ( F v ( t ) ) d t = 1 V R t ( d vol ( F v ( t ) ) ) = E N A ( ϕ F v ) (2.11) S L ( v ) := 1 V ∫ 0 + ∞   vol ⁡ F v ( t ) d t = 1 V ∫ R   t − d vol ⁡ F v ( t ) = E N A Ï• F v {:(2.11)S_(L)(v):=(1)/(V)int_(0)^(+oo)vol(F_(v)^((t)))dt=(1)/(V)int_(R)t(-d vol(F_(v)^((t))))=E^(NA)(phi_(F_(v))):}\begin{equation*} S_{L}(v):=\frac{1}{\mathbf{V}} \int_{0}^{+\infty} \operatorname{vol}\left(\mathscr{F}_{v}^{(t)}\right) d t=\frac{1}{\mathbf{V}} \int_{\mathbb{R}} t\left(-d \operatorname{vol}\left(\mathcal{F}_{v}^{(t)}\right)\right)=\mathbf{E}^{\mathrm{NA}}\left(\phi_{\mathscr{F}_{v}}\right) \tag{2.11} \end{equation*}(2.11)SL(v):=1V∫0+∞vol⁡(Fv(t))dt=1V∫Rt(−dvol⁡(Fv(t)))=ENA(Ï•Fv)
Let T ~ T ~ tilde(T)\tilde{\mathbb{T}}T~ be a maximal torus of Aut ( X ) Aut ⁡ ( X ) Aut(X)\operatorname{Aut}(X)Aut⁡(X) and ( X Q div ) T ~ X Q div  T ~ (X_(Q)^("div "))^( tilde(T))\left(X_{\mathbb{Q}}^{\text {div }}\right)^{\tilde{\mathbb{T}}}(XQdiv )T~ be the set of T ~ T ~ tilde(T)\tilde{\mathbb{T}}T~-invariant divisorial valuations. Define the following invariant ( ( ξ , v ) v ξ ( ξ , v ) ↦ v ξ ((xi,v)|->v_(xi):}\left((\xi, v) \mapsto v_{\xi}\right.((ξ,v)↦vξ is the action appeared in Example 2.8):
δ ( X ) = inf v X Q d i v A X ( v ) S X ( v ) , δ T ~ ( X ) = inf v ( X Q d i v ) sup ξ N ~ R A X ( v ξ ) S X ( v ξ ) δ ( X ) = inf v ∈ X Q d i v   A X ( v ) S X ( v ) , δ T ~ ( X ) = inf v ∈ X Q d i v   sup ξ ∈ N ~ R   A X v ξ S X v ξ delta(X)=i n f_(v inX_(Q)^(div))(A_(X)(v))/(S_(X)(v)),quaddelta_( tilde(T))(X)=i n f_(v in(X_(Q)^(div)))s u p_(xi in tilde(N)_(R))(A_(X)(v_(xi)))/(S_(X)(v_(xi)))\delta(X)=\inf _{v \in X_{\mathbb{Q}}^{\mathrm{div}}} \frac{A_{X}(v)}{S_{X}(v)}, \quad \delta_{\tilde{\mathbb{T}}}(X)=\inf _{v \in\left(X_{\mathbb{Q}}^{\mathrm{div}}\right)} \sup _{\xi \in \tilde{\mathbb{N}}_{\mathbb{R}}} \frac{A_{X}\left(v_{\xi}\right)}{S_{X}\left(v_{\xi}\right)}δ(X)=infv∈XQdivAX(v)SX(v),δT~(X)=infv∈(XQdiv)supξ∈N~RAX(vξ)SX(vξ)
Here we use the convention that A X ( v triv ) / S X ( v triv ) = + A X v triv  / S X v triv  = + ∞ A_(X)(v_("triv "))//S_(X)(v_("triv "))=+ooA_{X}\left(v_{\text {triv }}\right) / S_{X}\left(v_{\text {triv }}\right)=+\inftyAX(vtriv )/SX(vtriv )=+∞ for the trivial valuation v triv v triv  v_("triv ")v_{\text {triv }}vtriv .
Theorem 2.14. The following statements are true.
(1) ( [ 35 , 42 ] ) X ( [ 35 , 42 ] ) X ([35,42])X([35,42]) X([35,42])X is K K KKK-semistable if δ ( X ) 1 δ ( X ) ≥ 1 delta(X) >= 1\delta(X) \geq 1δ(X)≥1.
(2) ( [ 34 , 35 ] ) X ( [ 34 , 35 ] ) X ([34,35])X([34,35]) X([34,35])X is uniformly K K KKK-stable if and only if δ ( X ) > 1 δ ( X ) > 1 delta(X) > 1\delta(X)>1δ(X)>1.
(3) ( [ 15 , 35 , 42 ] ) X [ 15 , 35 , 42 ] ) X [15,35,42])X[15,35,42]) X[15,35,42])X is K K KKK-stable if and only if A X ( v ) > S ( v ) A X ( v ) > S ( v ) A_(X)(v) > S(v)A_{X}(v)>S(v)AX(v)>S(v) for any nontrivial v X Q div v ∈ X Q div  v inX_(Q)^("div ")v \in X_{\mathbb{Q}}^{\text {div }}v∈XQdiv .
(4) ([44]) X X XXX is reduced uniformly K K KKK-stable if and only if δ T ~ ( X ) > 1 δ T ~ ( X ) > 1 delta_( widetilde(T))(X) > 1\delta_{\widetilde{\mathbb{T}}}(X)>1δT~(X)>1.
To get these, we first use the fact as pointed out in [19] that for a special test configuration ( X , L ) ( X , L ) (X,L)(\mathcal{X}, \mathscr{L})(X,L), the valuation ord X 0 X 0 X_(0)\mathcal{X}_{0}X0 of the function field C ( X × C ) C ( X × C ) C(X xxC)\mathbb{C}(X \times \mathbb{C})C(X×C) restricts to become a divisorial valuation v X Q div v ∈ X Q div  v inX_(Q)^("div ")v \in X_{\mathbb{Q}}^{\text {div }}v∈XQdiv . A crucial observation is then made in [42]: F ( x , L ) λ R m = F ( x , L ) λ R m = F_((x,L))^(lambda)R_(m)=\mathcal{F}_{(x, \mathscr{L})}^{\lambda} R_{m}=F(x,L)λRm= F v λ + m A X ( v ) R m ( F v λ + m A X ( v ) R m F_(v)^(lambda+mA_(X)(v))R_(m)(:}\mathscr{F}_{v}^{\lambda+m A_{X}(v)} R_{m}\left(\right.Fvλ+mAX(v)Rm( see (2.9)). This implies vol ( F ( X , L ) ( t ) ) = vol ( F v ( t + A X ( v ) ) ) vol ⁡ F ( X , L ) ( t ) = vol ⁡ F v t + A X ( v ) vol(F_((X,L))^((t)))=vol(F_(v)^((t+A_(X)(v))))\operatorname{vol}\left(\mathscr{F}_{(X, \mathscr{L})}^{(t)}\right)=\operatorname{vol}\left(\mathcal{F}_{v}^{\left(t+A_{X}(v)\right)}\right)vol⁡(F(X,L)(t))=vol⁡(Fv(t+AX(v))), which, together with (2.10), leads to 2 2 ^(2){ }^{2}2
(2.12) M N A ( X , L ) = D N A ( X , L ) = A X ( v ) E N A ( F v ) = A X ( v ) S ( v ) (2.12) M N A ( X , L ) = D N A ( X , L ) = A X ( v ) − E N A F v = A X ( v ) − S ( v ) {:(2.12)M^(NA)(X","L)=D^(NA)(X","L)=A_(X)(v)-E^(NA)(F_(v))=A_(X)(v)-S(v):}\begin{equation*} \mathbf{M}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L})=\mathbf{D}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L})=A_{X}(v)-\mathbf{E}^{\mathrm{NA}}\left(\mathcal{F}_{v}\right)=A_{X}(v)-S(v) \tag{2.12} \end{equation*}(2.12)MNA(X,L)=DNA(X,L)=AX(v)−ENA(Fv)=AX(v)−S(v)
This, together with Theorem 2.13, already gives the sufficient condition for the K-(semi)stability. The criterion for uniform K-stability follows from a similar argument and K. Fujita's inequality, 1 n S ( v ) J N A ( F v ) n S ( v ) 1 n S ( v ) ≤ J N A F v ≤ n S ( v ) (1)/(n)S(v) <= J^(NA)(F_(v)) <= nS(v)\frac{1}{n} S(v) \leq \mathbf{J}^{\mathrm{NA}}\left(\mathcal{F}_{v}\right) \leq n S(v)1nS(v)≤JNA(Fv)≤nS(v) [34]. For reduced uniform stability, another identity A X ( v ξ ) S ( v ξ ) = A X ( v ) S ( v ) + A X v ξ − S v ξ = A X ( v ) − S ( v ) + A_(X)(v_(xi))-S(v_(xi))=A_(X)(v)-S(v)+A_{X}\left(v_{\xi}\right)-S\left(v_{\xi}\right)=A_{X}(v)-S(v)+AX(vξ)−S(vξ)=AX(v)−S(v)+ Fut X ( ξ ) X ( ξ ) _(X)(xi)_{X}(\xi)X(ξ) proved in [44] is needed.
As we will see in Section 3.2, a main reason for introducing the (reduced) uniform K-stability is that it is much easier to use in making connection with the (reduced) coercivity in the complex analytic setting. On the other hand, we now have the following fundamental result:
Theorem 2.15 ([56]). Let X X XXX be a Q Q Q\mathbb{Q}Q-Fano variety. Then X X XXX is K K KKK-stable if and only if X X XXX is uniformly K K KKK-stable. More generally, X X XXX is reduced uniformly stable if and only if X X XXX is K K KKK polystable. Moreover, these statements hold true for any log Fano pair.
This is achieved by several works. First, according to a work of Blum-Liu-Xu [13], divisorial valuations on X X XXX associated to special test configurations are log canonical places of complements. By deep boundedness of Birkar and Haccon-McKernan-Xu, it was also shown that there exists a quasimonomial valuation (i.e., a monomial valuation on a smooth birational model) that achieves the infimum defining δ ( X ) δ ( X ) delta(X)\delta(X)δ(X) (and a similar result holds more generally for δ T ~ ( X ) ) δ T ~ ( X ) {:delta_( tilde(T))(X))\left.\delta_{\tilde{\mathbb{T}}}(X)\right)δT~(X)). Then the main problem becomes proving a finite generation property for the minimizing valuation, which is achieved by using deep techniques from birational
2 The original argument in [42] also explicitly relates the filtration F ( X , L ) F ( X , L {:F_(()X,L)\left.\mathcal{F}_{(} \mathcal{X}, \mathscr{L}\right)F(X,L) to a filtration of the section ring of X 0 X 0 X_(0)\mathcal{X}_{0}X0 induced by the C C ∗ C^(**)\mathbb{C}^{*}C∗-action.
algebraic geometry in [56]. In fact, in the past several years, the algebraic study of K-stability for Fano varieties has flourished, and there are many important results which answer fundamental questions in this subject. We highlight two such achievements:
(1) Algebraic construction of projective moduli space of K-polystable Fano varieties. This is achieved in a collection of works, settling different issues in the construction including boundedness, separatedness, properness, and projectivity. Moreover, concrete examples of compact moduli spaces have been identified. We refer to [ 56 , 70 ] [ 56 , 70 ] [56,70][56,70][56,70] for extensive discussions on related topics.
(2) Fujita-Odaka [36] introduced quantizations of the δ ( X ) δ ( X ) delta(X)\delta(X)δ(X) invariant: for each m N m ∈ N m inNm \in \mathbb{N}m∈N,
δ m ( X ) = inf { lct ( X , D ) : D is of m -basis type } δ m ( X ) = inf { lct ⁡ ( X , D ) : D  is of  m -basis type  } delta_(m)(X)=i n f{lct(X,D):D" is of "m"-basis type "}\delta_{m}(X)=\inf \{\operatorname{lct}(X, D): D \text { is of } m \text {-basis type }\}δm(X)=inf{lct⁡(X,D):D is of m-basis type }
where D D DDD is of m m mmm-basis type if D = 1 m N m i = 1 N m { s i = 0 } D = 1 m N m ∑ i = 1 N m   s i = 0 D=(1)/(mN_(m))sum_(i=1)^(N_(m)){s_(i)=0}D=\frac{1}{m N_{m}} \sum_{i=1}^{N_{m}}\left\{s_{i}=0\right\}D=1mNm∑i=1Nm{si=0} where { s i } s i {s_(i)}\left\{s_{i}\right\}{si} is a basis of H 0 ( X , m L ) H 0 ( X , m L ) H^(0)(X,mL)H^{0}(X, m L)H0(X,mL). Blum-Jonsson [12] proved lim m + δ m ( X ) = δ ( X ) lim m → + ∞   δ m ( X ) = δ ( X ) lim_(m rarr+oo)delta_(m)(X)=delta(X)\lim _{m \rightarrow+\infty} \delta_{m}(X)=\delta(X)limm→+∞δm(X)=δ(X). This provides a practical tool to verify uniform stability of Fano varieties. Ahmadinezhad-Zhuang [1] further introduced new techniques for estimating the δ m δ m delta_(m)\delta_{m}δm and δ δ delta\deltaδ invariant which lead to many new examples of K-stable Fano varieties. All of these culminate in the recent determination of deformation types of smooth Fano threefolds that contain K-polystable ones (see [3]).
In another direction, Han-Li [37] establishes a valuative criterion for g g ggg-weighted stability, corresponding to the study of g g ggg-solitons. A key idea in such an extension is using a fibration technique for a polynomial weights (as motivated by the theory of equivariant de Rham cohomology) and then using the Stone-Weierstrass approximation to deal with the general g g ggg. Moreover, there is a notion of stability for Fano cones introduced earlier by Collins-Székelyhidi associated to Ricci-flat Kähler cone metrics. It is shown recently that this stability of Fano cones is, in fact, equivalent to a particular g g ggg-weighted stability of log Fano quotients (see [ 2 , 47 ] [ 2 , 47 ] [2,47][2,47][2,47] ).
The techniques developed in the study of (weighted) K-stability of Fano varieties have also been applied to treat an optimal degeneration problem that is motivated by the Hamilton-Tian conjecture in differential geometry (see [74] for background of this conjecture). This is formulated as a minimization problem for valuations in [38] which defines the following functional (cf. (2.12) and (2.11)), for any valuation v X N A v ∈ X N A v inX^(NA)v \in X^{\mathrm{NA}}v∈XNA,
β ~ ( v ) = A X ( v ) + log ( 1 V 0 + e t ( d vol ( F v ( t ) ) ) ) β ~ ( v ) = A X ( v ) + log ⁡ 1 V ∫ 0 + ∞   e − t − d vol ⁡ F v ( t ) tilde(beta)(v)=A_(X)(v)+log((1)/(V)int_(0)^(+oo)e^(-t)(-d vol(F_(v)^((t)))))\tilde{\beta}(v)=A_{X}(v)+\log \left(\frac{1}{\mathbf{V}} \int_{0}^{+\infty} e^{-t}\left(-d \operatorname{vol}\left(\mathscr{F}_{v}^{(t)}\right)\right)\right)β~(v)=AX(v)+log⁡(1V∫0+∞e−t(−dvol⁡(Fv(t))))
Very roughly speaking, the β ~ β ~ tilde(beta)\tilde{\beta}β~ functional is an antiderivative of a certain weighted Futaki invariant. This functional is a variant of invariants that appeared in previous works of TianZhang-Zhang-Zhu, Dervan-Székelyhidi, and Hisamoto (see [74] for more details). The results from [ 14 , 37 , 51 ] [ 14 , 37 , 51 ] [14,37,51][14,37,51][14,37,51] together prove the following algebraic version of Hamilton-Tian conjecture:
Theorem 2.16. For any Q Q Q\mathbb{Q}Q-Fano variety, there exists a unique quasimonomial valuation v v ∗ v_(**)v_{*}v∗ that minimizes β ~ β ~ tilde(beta)\tilde{\beta}β~, whose associated filtration F v F v ∗ F_(v_(**))\mathcal{F}_{v_{*}}Fv∗ is finitely generated and induces a degener-
ation of X X XXX to a Q Q Q\mathbb{Q}Q-Fano variety X 0 X 0 X_(0)X_{0}X0 together with a vector field V ξ V ξ V_(xi)V_{\xi}Vξ. Moreover, X 0 X 0 X_(0)X_{0}X0 degenerates uniquely to an e , ξ e − ⟨ â‹… , ξ ⟩ e^(-(:*,xi:))e^{-\langle\cdot, \xi\rangle}e−⟨⋅,ξ⟩-weighted polystable Q Q Q\mathbb{Q}Q-Fano variety (cf. Example 1.1).
Combined with previous works, the uniqueness part in particular confirms a conjecture of Chen-Sun-Wang about the algebraic uniqueness of limits under normalized KählerRicci flows on Fano manifolds (see [62]).

2.4. Normalized volume and local stability theory of klt singularities

A similar minimization problem for valuations was actually studied even earlier in the local setting, which motivates the formulation and the proof of Theorem 2.16. Let ( X , x ) ( X , x ) (X,x)(X, x)(X,x) be a klt singularity. Denote by V a l X , x V a l X , x Val_(X,x)\mathrm{Val}_{X, x}ValX,x the space of real valuations that have center x x xxx. The following normalized volume functional was introduced in [43]: for any v Val X , x v ∈ Val X , x v inVal_(X,x)v \in \operatorname{Val}_{X, x}v∈ValX,x,
(2.13) vol ^ ( v ) := { A X ( v ) n vol ( v ) , if A X ( v ) < + + , otherwise (2.13) vol ^ ( v ) := A X ( v ) n ⋅ vol ⁡ ( v ) ,  if  A X ( v ) < + ∞ + ∞ ,  otherwise  {:(2.13) widehat(vol)(v):={[A_(X)(v)^(n)*vol(v)","," if "A_(X)(v) < +oo],[+oo","," otherwise "]:}:}\widehat{\operatorname{vol}}(v):= \begin{cases}A_{X}(v)^{n} \cdot \operatorname{vol}(v), & \text { if } A_{X}(v)<+\infty \tag{2.13}\\ +\infty, & \text { otherwise }\end{cases}(2.13)vol^(v):={AX(v)n⋅vol⁡(v), if AX(v)<+∞+∞, otherwise 
Here A X ( v ) A X ( v ) A_(X)(v)A_{X}(v)AX(v) is again the log log log\loglog discrepancy functional and vol ( v ) vol ⁡ ( v ) vol(v)\operatorname{vol}(v)vol⁡(v) is defined as
vol ( v ) = lim p + dim C ( O X , x / a p ( v ) ) p n / n ! where a p ( v ) = { f O X , x ; v ( f ) p } vol ⁡ ( v ) = lim p → + ∞   dim C ⁡ O X , x / a p ( v ) p n / n !  where  a p ( v ) = f ∈ O X , x ; v ( f ) ≥ p vol(v)=lim_(p rarr+oo)(dim_(C)(O_(X,x)//a_(p)(v)))/(p^(n)//n!)quad" where "a_(p)(v)={f inO_(X,x);v(f) >= p}\operatorname{vol}(v)=\lim _{p \rightarrow+\infty} \frac{\operatorname{dim}_{\mathbb{C}}\left(\mathcal{O}_{X, x} / a_{p}(v)\right)}{p^{n} / n!} \quad \text { where } a_{p}(v)=\left\{f \in \mathcal{O}_{X, x} ; v(f) \geq p\right\}vol⁡(v)=limp→+∞dimC⁡(OX,x/ap(v))pn/n! where ap(v)={f∈OX,x;v(f)≥p}
The expression in (2.13) is inspired by the work of Martelli-Sparks-Yau [57] on a volume minimization property of Reeb vector fields associated to Ricci-flat Kähler cone metrics. In [43] we started to consider the minimization of vol over ^ Val X , x  vol over  ^ Val X , x widehat(" vol over ")Val_(X,x)\widehat{\text { vol over }} \operatorname{Val}_{X, x} vol over ^ValX,x and define the invariant vol ^ ( X , x ) = inf v V a l X , x vol ^ ( v ) vol ^ ( X , x ) = inf v ∈ V a l X , x   vol ^ ( v ) widehat(vol)(X,x)=i n f_(v inVal_(X,x)) widehat(vol)(v)\widehat{\operatorname{vol}}(X, x)=\inf _{v \in \mathrm{Val}_{X, x}} \widehat{\operatorname{vol}}(v)vol^(X,x)=infv∈ValX,xvol^(v). We proved that the invariant vol ^ ( X , x ) vol ^ ( X , x ) widehat(vol)(X,x)\widehat{\operatorname{vol}}(X, x)vol^(X,x) is strictly positive and further conjectured the existence, uniqueness of minimizing valuations which should have finite generated associated graded rings. For a concrete example, it was shown by the author and Y. Liu that for an isolated quotient singularity X = C n / Γ , vol ^ ( C n / Γ , 0 ) = n n | Γ | X = C n / Γ , vol ^ C n / Γ , 0 = n n | Γ | X=C^(n)//Gamma, widehat(vol)(C^(n)//Gamma,0)=(n^(n))/(|Gamma|)X=\mathbb{C}^{n} / \Gamma, \widehat{\operatorname{vol}}\left(\mathbb{C}^{n} / \Gamma, 0\right)=\frac{n^{n}}{|\Gamma|}X=Cn/Γ,vol^(Cn/Γ,0)=nn|Γ| and the exceptional divisor of the standard blowup obtains the infimum.
This minimization problem was proposed by the author to attack a conjecture of Donaldson-Sun, which states that the metric tangent cone at any point on a GromovHausdorff limit of Kähler-Einstein manifolds depends only on the algebraic structure (see [62]). This conjecture has been confirmed in a series of following-up papers [51, 53, 54]. Algebraically, we have the following results regarding this minimization problem.
Theorem 2.17. (1) There exists a valuation that achieves the infimum in defining vol ^ ( X , x ) vol ^ ( X , x ) widehat(vol)(X,x)\widehat{\operatorname{vol}}(X, x)vol^(X,x). Moreover, this minimizing valuation is quasimonomial and unique up to rescaling.
(2) A divisorial valuation v v ∗ v_(**)v_{*}v∗ is the minimizer if and only if it is the exceptional divisor of a plt blowup and also the associated log Fano pair is K K KKK-semistable.
The first statement is a combination of works by Harold Blum, Chenyang Xu, and Ziquan Zhuang [11,71,72] partly based on calculations from [42,43]. The second statement was proved in L i X u L i − X u Li-Xu\mathrm{Li}-\mathrm{Xu}Li−Xu [54] (see also [11]) by extending the global argument from [52] to the local case, and it shows a close relationship between the local and global theory. In fact, it is in proving the affine cone case of this statement when valuative criterion for K K K\mathrm{K}K-(semi)stability
was first discovered in [42]. A similar statement is true for more general quasimonomial minimizing valuations [53]. However, the finite generation conjecture from [43] is still open in general, and seems to require deeper boundedness property of Fano varieties.
We also like to mention that Yuchen Liu obtained a surprising local-to-global comparison inequality by generalizing an estimate of Kento Fujita:
Theorem 2.18 ([55]). For any closed point x x xxx on a K K KKK-semistable Q Q Q\mathbb{Q}Q-Fano variety X X XXX, we have
(2.14) ( K X ) n ( n + 1 ) n n n vol ^ ( X , x ) (2.14) − K X ⋅ n ≤ ( n + 1 ) n n n vol ^ ( X , x ) {:(2.14)(-K_(X))^(*n) <= ((n+1)^(n))/(n^(n)) widehat(vol)(X","x):}\begin{equation*} \left(-K_{X}\right)^{\cdot n} \leq \frac{(n+1)^{n}}{n^{n}} \widehat{\operatorname{vol}}(X, x) \tag{2.14} \end{equation*}(2.14)(−KX)⋅n≤(n+1)nnnvol^(X,x)
For example, if x X x ∈ X x in Xx \in Xx∈X is a regular point, (2.14) recovers Fujita's beautiful inequality, namely ( K X ) n ( n + 1 ) n − K X ⋅ n ≤ ( n + 1 ) n (-K_(X))^(*n) <= (n+1)^(n)\left(-K_{X}\right)^{\cdot n} \leq(n+1)^{n}(−KX)⋅n≤(n+1)n for any K-semistable X X XXX [33]. Inequality (2.14) has applications in controlling singularities on the varieties that correspond to boundary points of moduli spaces. In order for this to be effective, good estimates of vol ^ ( X , x ) vol ^ ( X , x ) widehat(vol)(X,x)\widehat{\operatorname{vol}}(X, x)vol^(X,x) for klt singularities need to be developed. In particular, it is still interesting to understand better the vol invariants and ^  vol invariants and  ^ widehat(" vol invariants and ")\widehat{\text { vol invariants and }} vol invariants and ^ associated minimizers for 3-dimensional klt singularities. For more discussion on related topics, we refer to the survey [48].

3. ARCHIMEDEAN (COMPLEX ANALYTIC) THEORY VS. NON-ARCHIMEDEAN THEORY

3.1. Correspondence between Archimedean and non-Archimedean objects

In this section, we discuss results showing a general philosophy that non-Archimedean objects encode the information of corresponding Archimedean objects "at infinity."
Let ( X , L ) ( X , L ) (X,L)(\mathcal{X}, \mathscr{L})(X,L) be a test configuration and h ~ h ~ tilde(h)\tilde{h}h~ be a smooth psh metric on L L L\mathscr{L}L. Via the isomorphism ( X , L ) × C C X × C ( X , L ) × C C ∗ ≅ X × C ∗ (X,L)xx_(C)C^(**)~=X xxC^(**)(\mathcal{X}, \mathscr{L}) \times_{\mathbb{C}} \mathbb{C}^{*} \cong X \times \mathbb{C}^{*}(X,L)×CC∗≅X×C∗, we get a path Φ ~ = { φ ~ ( s ) } s R Φ ~ = { φ ~ ( s ) } s ∈ R widetilde(Phi)={ tilde(varphi)(s)}_(s inR)\widetilde{\Phi}=\{\tilde{\varphi}(s)\}_{s \in \mathbb{R}}Φ~={φ~(s)}s∈R of smooth ω 0 ω 0 omega_(0)\omega_{0}ω0-psh potentials where s = log | t | 2 s = − log ⁡ | t | 2 s=-log |t|^(2)s=-\log |t|^{2}s=−log⁡|t|2. With these notation, we have the following slope formula:
Theorem 3.1 ([20,59,64,67]). The slope at infinity of a functional F { E , Λ , I , J , M } F ∈ { E , Λ , I , J , M } Fin{E,Lambda,I,J,M}\mathbf{F} \in\{\mathbf{E}, \boldsymbol{\Lambda}, \mathbf{I}, \mathbf{J}, \mathbf{M}\}F∈{E,Λ,I,J,M} is given by the corresponding non-Archimedean functional
F ( Φ ~ ) := lim s + F ( φ ~ ( s ) ) s = F N A ( X , L ) = F N A ( ϕ ( x , L ) ) F ′ ∞ ( Φ ~ ) := lim s → + ∞   F ( φ ~ ( s ) ) s = F N A ( X , L ) = F N A Ï• ( x , L ) F^('oo)( widetilde(Phi)):=lim_(s rarr+oo)(F(( tilde(varphi))(s)))/(s)=F^(NA)(X,L)=F^(NA)(phi_((x,L)))\mathbf{F}^{\prime \infty}(\widetilde{\Phi}):=\lim _{s \rightarrow+\infty} \frac{\mathbf{F}(\tilde{\varphi}(s))}{s}=\mathbf{F}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L})=\mathbf{F}^{\mathrm{NA}}\left(\phi_{(x, \mathscr{L})}\right)F′∞(Φ~):=lims→+∞F(φ~(s))s=FNA(X,L)=FNA(Ï•(x,L))
There is a more canonical analytic object associated to a test configuration. Recall from section 1.5 that by a geodesic ray Φ = { φ ( s ) } s [ 0 , + ) Φ = { φ ( s ) } s ∈ [ 0 , + ∞ ) Phi={varphi(s)}_(s in[0,+oo))\Phi=\{\varphi(s)\}_{s \in[0,+\infty)}Φ={φ(s)}s∈[0,+∞) in E 1 E 1 E^(1)\mathcal{E}^{1}E1 we mean that Φ | [ s 1 , s 2 ] Φ s 1 , s 2 Phi|_([s_(1),s_(2)])\left.\Phi\right|_{\left[s_{1}, s_{2}\right]}Φ|[s1,s2] is a geodesic connecting φ ( s 1 ) , φ ( s 2 ) φ s 1 , φ s 2 varphi(s_(1)),varphi(s_(2))\varphi\left(s_{1}\right), \varphi\left(s_{2}\right)φ(s1),φ(s2) for any s 1 , s 2 [ 0 , ) s 1 , s 2 ∈ [ 0 , ∞ ) s_(1),s_(2)in[0,oo)s_{1}, s_{2} \in[0, \infty)s1,s2∈[0,∞) (see (1.12)).
Theorem 3.2 ([60]). Given any test configuration ( X , L ) ( X , L ) (X,L)(\mathcal{X}, \mathscr{L})(X,L) for ( X , L ) ( X , L ) (X,L)(X, L)(X,L), there exists a geodesic ray Φ ( X , L ) Φ ( X , L ) Phi_((X,L))\Phi_{(X, \mathscr{L})}Φ(X,L) emanating from any given smooth potential φ 0 φ 0 varphi_(0)\varphi_{0}φ0.
On the other hand, recall from section 2.1 that there is a non-Archimedean potential associated to ( X , L ) ( X , L ) (X,L)(\mathcal{X}, \mathscr{L})(X,L) (see (2.1)). Berman-Boucksom-Jonsson proved that there is a direct relation between geodesic rays and non-Archimedean potentials. First they showed that any geodesic ray Φ Î¦ Phi\PhiΦ defines a non-Archimedean potential (cf. (2.1))
Φ N A ( v ) := G ( v ) ( Φ ) , for any v X Q d i v Φ N A ( v ) := − G ( v ) ( Φ ) ,  for any  v ∈ X Q d i v Phi_(NA)(v):=-G(v)(Phi),quad" for any "v inX_(Q)^(div)\Phi_{\mathrm{NA}}(v):=-G(v)(\Phi), \quad \text { for any } v \in X_{\mathbb{Q}}^{\mathrm{div}}ΦNA(v):=−G(v)(Φ), for any v∈XQdiv
where G ( v ) ( Φ ) G ( v ) ( Φ ) G(v)(Phi)G(v)(\Phi)G(v)(Φ) is the generic Lelong number of Φ Î¦ Phi\PhiΦ considered as a singular quasi-psh potential on a birational model where the center of the valuation G ( v ) G ( v ) G(v)G(v)G(v) is a prime divisor.
Theorem 3.3 ([7]). The following statements are true:
(1) The map Φ Φ N A Φ ↦ Φ N A Phi|->Phi_(NA)\Phi \mapsto \Phi_{\mathrm{NA}}Φ↦ΦNA has the image contained in ( E 1 ) N A E 1 N A (E^(1))^(NA)\left(\mathcal{E}^{1}\right)^{\mathrm{NA}}(E1)NA. Conversely, for any ϕ Ï• ∈ phi in\phi \inϕ∈ ( ε 1 ) N A ε 1 N A (epsi^(1))^(NA)\left(\varepsilon^{1}\right)^{\mathrm{NA}}(ε1)NA, there exists a geodesic ray denoted by γ ( ϕ ) γ ( Ï• ) gamma(phi)\gamma(\phi)γ(Ï•) that satisfies γ ( ϕ ) N A = ϕ γ ( Ï• ) N A = Ï• gamma(phi)_(NA)=phi\gamma(\phi)_{\mathrm{NA}}=\phiγ(Ï•)NA=Ï•.
(2) For any geodesic ray Φ , Φ ^ = γ ( Φ N A ) Φ , Φ ^ = γ Φ N A Phi, widehat(Phi)=gamma(Phi_(NA))\Phi, \widehat{\Phi}=\gamma\left(\Phi_{\mathrm{NA}}\right)Φ,Φ^=γ(ΦNA) satisfies Φ ^ N A = Φ N A ( E 1 ) N A Φ ^ N A = Φ N A ∈ E 1 N A widehat(Phi)_(NA)=Phi_(NA)in(E^(1))^(NA)\widehat{\Phi}_{\mathrm{NA}}=\Phi_{\mathrm{NA}} \in\left(\mathcal{E}^{1}\right)^{\mathrm{NA}}Φ^NA=ΦNA∈(E1)NA and Φ ^ Φ Î¦ ^ ≥ Φ widehat(Phi) >= Phi\widehat{\Phi} \geq \PhiΦ^≥Φ.
(3) For Φ = γ ( ϕ ) Φ = γ ( Ï• ) Phi=gamma(phi)\Phi=\gamma(\phi)Φ=γ(Ï•) with ϕ ( ε 1 ) N A , E ( Φ ) = E N A ( ϕ ) Ï• ∈ ε 1 N A , E ′ ∞ ( Φ ) = E N A ( Ï• ) phi in(epsi^(1))^(NA),E^('oo)(Phi)=E^(NA)(phi)\phi \in\left(\varepsilon^{1}\right)^{\mathrm{NA}}, \mathbf{E}^{\prime \infty}(\Phi)=\mathbf{E}^{\mathrm{NA}}(\phi)ϕ∈(ε1)NA,E′∞(Φ)=ENA(Ï•), and there exists a sequence of test configurations ( X m , L m ) X m , L m (X_(m),L_(m))\left(\mathcal{X}_{m}, \mathscr{L}_{m}\right)(Xm,Lm) such that Φ Î¦ Phi\PhiΦ is the decreasing limit of Φ ( X m , L m ) Φ X m , L m Phi_((X_(m),L_(m)))\Phi_{\left(X_{m}, \mathscr{L}_{m}\right)}Φ(Xm,Lm) (see Theorem 3.2).
Berman-Boucksom-Jonsson proved Φ N A P S H N A Φ N A ∈ P S H N A Phi_(NA)inPSH^(NA)\Phi_{\mathrm{NA}} \in \mathrm{PSH}^{\mathrm{NA}}ΦNA∈PSHNA by blowing up multiplier ideal sheaves { L ( m Φ ) } m N { L ( m Φ ) } m ∈ N {L(m Phi)}_(m inN)\{\mathcal{L}(m \Phi)\}_{m \in \mathbb{N}}{L(mΦ)}m∈N and using their global generation properties to construct test configurations { ( X m , L m ) } X m , L m {(X_(m),L_(m))}\left\{\left(\mathcal{X}_{m}, \mathscr{L}_{m}\right)\right\}{(Xm,Lm)} such that ϕ ( X m , L m ) Ï• X m , L m phi_((X_(m),L_(m)))\phi_{\left(X_{m}, \mathscr{L}_{m}\right)}Ï•(Xm,Lm) decreases to Φ N A Φ N A Phi_(NA)\Phi_{\mathrm{NA}}ΦNA. Because of the second statement, any geodesic ray γ ( ϕ ) γ ( Ï• ) gamma(phi)\gamma(\phi)γ(Ï•) with ϕ ( E 1 ) N A Ï• ∈ E 1 N A phi in(E^(1))^(NA)\phi \in\left(\mathcal{E}^{1}\right)^{\mathrm{NA}}ϕ∈(E1)NA is called maximal in [7]. By the last statement, maximal geodesic rays can be approximated by (geodesic rays associated to) test configurations. Moreover, when ϕ = ϕ ( x , L ) H N A , γ ( ϕ ) Ï• = Ï• ( x , L ) ∈ H N A , γ ( Ï• ) phi=phi_((x,L))inH^(NA),gamma(phi)\phi=\phi_{(x, \mathscr{L})} \in \mathscr{H}^{\mathrm{NA}}, \gamma(\phi)Ï•=Ï•(x,L)∈HNA,γ(Ï•) coincides with the geodesic ray from Theorem 3.2. Further useful properties of maximal geodesic rays are known (cf. Theorem 3.1):
Theorem 3.4 ([45]). Let Φ Î¦ Phi\PhiΦ be a maximal geodesic ray.
(1) We have the identity ( E Ric ( ω 0 ) ) ( Φ ) = ( E K ) N A ( Φ N A ) E − Ric ⁡ ω 0 ′ ∞ ( Φ ) = E K N A Φ N A (E^(-Ric(omega_(0))))^('oo)(Phi)=(E^(K))^(NA)(Phi_(NA))\left(\mathbf{E}^{-\operatorname{Ric}\left(\omega_{0}\right)}\right)^{\prime \infty}(\Phi)=\left(\mathbf{E}^{K}\right)^{\mathrm{NA}}\left(\Phi_{\mathrm{NA}}\right)(E−Ric⁡(ω0))′∞(Φ)=(EK)NA(ΦNA).
(2) H ( Φ ) H N A ( Φ N A ) H ′ ∞ ( Φ ) ≥ H N A Φ N A H^('oo)(Phi) >= H^(NA)(Phi_(NA))\mathbf{H}^{\prime \infty}(\Phi) \geq \mathbf{H}^{\mathrm{NA}}\left(\Phi_{\mathrm{NA}}\right)H′∞(Φ)≥HNA(ΦNA). Moreover, if Φ = Φ ( x , L ) Φ = Φ ( x , L ) Phi=Phi_((x,L))\Phi=\Phi_{(x, \mathscr{L})}Φ=Φ(x,L) is associated to a test configuration, then H ( Φ ) = H N A ( Φ N A ) H ′ ∞ ( Φ ) = H N A Φ N A H^('oo)(Phi)=H^(NA)(Phi_(NA))\mathbf{H}^{\prime \infty}(\Phi)=\mathbf{H}^{\mathrm{NA}}\left(\Phi_{\mathrm{NA}}\right)H′∞(Φ)=HNA(ΦNA).
It is natural to conjecture that H ( Φ ) = H N A ( Φ N A ) H ′ ∞ ( Φ ) = H N A Φ N A H^('oo)(Phi)=H^(NA)(Phi_(NA))\mathbf{H}^{\prime \infty}(\Phi)=\mathbf{H}^{\mathrm{NA}}\left(\Phi_{\mathrm{NA}}\right)H′∞(Φ)=HNA(ΦNA) always holds for any maximal geodesic ray Φ Î¦ Phi\PhiΦ. This is implied by the algebraic Conjecture 2.12, according to [45, 46].
As pointed out in [7], by a construction of Darvas, there are abundant nonmaximal geodesic rays. In fact, analogous local examples have been used by the author to disprove a conjecture of Demailly on Monge-Ampère mass of psh singularities. It is thus a surprising fact that maximal geodesic rays are the only ones of interest in the c s c K c s c K cscK\mathrm{cscK}cscK problem.
Theorem 3.5 ([45]). If a geodesic ray Φ Î¦ Phi\PhiΦ satisfies M ( Φ ) < + M ′ ∞ ( Φ ) < + ∞ M^('oo)(Phi) < +oo\mathbf{M}^{\prime \infty}(\Phi)<+\inftyM′∞(Φ)<+∞, then Φ Î¦ Phi\PhiΦ is maximal.
Note that M ( Φ ) = lim s + M ( φ ( s ) ) s M ′ ∞ ( Φ ) = lim s → + ∞   M ( φ ( s ) ) s M^('oo)(Phi)=lim_(s rarr+oo)(M(varphi(s)))/(s)\mathbf{M}^{\prime \infty}(\Phi)=\lim _{s \rightarrow+\infty} \frac{\mathbf{M}(\varphi(s))}{s}M′∞(Φ)=lims→+∞M(φ(s))s exists by Theorem 1.3. This result resolves a difficulty raised in Boucksom's ICM talk [16], and implies that destabilizing geodesic rays can always be approximated by test configurations, thus giving a very strong evidence for the validity of Yau-Tian-Donaldson Conjecture 3.6. The proof of Theorem 3.5 starts with an equisingularity property X × { | t | < 1 } e α ( Φ ^ Φ ) < + ∫ X × { | t | < 1 }   e − α ( Φ ^ − Φ ) < + ∞ int_(X xx{|t| < 1})e^(-alpha( widehat(Phi)-Phi)) < +oo\int_{X \times\{|t|<1\}} e^{-\alpha(\widehat{\Phi}-\Phi)}<+\infty∫X×{|t|<1}e−α(Φ^−Φ)<+∞ for any α > 0 α > 0 alpha > 0\alpha>0α>0, and then uses Jensen's inequality, together with a comparison principle, for the E E E\mathbf{E}E functional to get a contradiction with the finite slope assumption if Φ ^ = γ ( Φ N A ) Φ Î¦ ^ = γ Φ N A ≠ Φ widehat(Phi)=gamma(Phi_(NA))!=Phi\widehat{\Phi}=\gamma\left(\Phi_{\mathrm{NA}}\right) \neq \PhiΦ^=γ(ΦNA)≠Φ.

3.2. Yau-Tian-Donaldson conjecture for general polarized manifolds

The Yau-Tian-Donaldson (YTD) conjecture says that the existence of canonical Kähler metrics on projective manifolds should be equivalent to a certain K-stability condition. For a general polarization, it is believed that one needs to use a strengthened definition of K-stability such as Definition 2.3. In particular, we have the following version.
Conjecture 3.6 (uniform YTD conjecture). A polarized manifold ( X , L ) ( X , L ) (X,L)(X, L)(X,L) admits a cscK metric if and only if ( X , L ) ( X , L ) (X,L)(X, L)(X,L) is reduced uniformly K K KKK-stable.
The implication from existence to stability is known, and follows from Theorems 1.4 and 3.1. The other direction is still open in general. However, based on the results discussed thus far, we can explain the proof of a weak version.
Theorem 3.7 ([45]). If ( X , L ) ( X , L ) (X,L)(X, L)(X,L) is uniformly stable over models (i.e., there exists γ > 0 γ > 0 gamma > 0\gamma>0γ>0 such that M N A ( X , L ) γ J N A ( X , L ) M N A ( X , L ) ≥ γ â‹… J N A ( X , L ) M^(NA)(X,L) >= gamma*J^(NA)(X,L)\mathbf{M}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L}) \geq \gamma \cdot \mathbf{J}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L})MNA(X,L)≥γ⋅JNA(X,L) for any model ( X , L ) ) ( X , L ) {:(X,L))\left.(\mathcal{X}, \mathscr{L})\right)(X,L)), then it admits a cscK metric.
Summary of proof. Step 1. By Theorem 1.4, we need to show that M M M\mathbf{M}M is coercive. Assume that the coercivity fails. Then there exists a geodesic ray Φ = { φ ( s ) } s [ 0 , ) Φ = { φ ( s ) } s ∈ [ 0 , ∞ ) Phi={varphi(s)}_(s in[0,oo))\Phi=\{\varphi(s)\}_{s \in[0, \infty)}Φ={φ(s)}s∈[0,∞) satisfying
M ( Φ ) 0 , J ( Φ ) = 1 , sup ( φ ( s ) ) = 0 M ′ ∞ ( Φ ) ≤ 0 , J ′ ∞ ( Φ ) = 1 , sup ( φ ( s ) ) = 0 M^('oo)(Phi) <= 0,quadJ^('oo)(Phi)=1,quad s u p(varphi(s))=0\mathbf{M}^{\prime \infty}(\Phi) \leq 0, \quad \mathbf{J}^{\prime \infty}(\Phi)=1, \quad \sup (\varphi(s))=0M′∞(Φ)≤0,J′∞(Φ)=1,sup(φ(s))=0
Such a destabilizing geodesic ray Φ Î¦ Phi\PhiΦ was constructed in [ 7 , 27 ] [ 7 , 27 ] [7,27][7,27][7,27] from a destabilizing sequence. In this construction, both the convexity of M M M\mathbf{M}M from Theorem 1.3 and a compactness result for potentials with uniform entropy bounds from [6] play crucial roles.
Step 2. By Theorem 3.5, Φ Î¦ Phi\PhiΦ is maximal. Set ϕ = Φ N A Ï• = Φ N A phi=Phi_(NA)\phi=\Phi_{\mathrm{NA}}Ï•=ΦNA. By using Theorem 3.3(3) and Theorem 3.4(1), we derive the identities
E ( Φ ) = E N A ( ϕ ) , ( E Ric ( ω 0 ) ) ( Φ ) = ( E K X ) N A ( ϕ ) , J ( Φ ) = J N A ( ϕ ) E ′ ∞ ( Φ ) = E N A ( Ï• ) , E − Ric ⁡ ω 0 ′ ∞ ( Φ ) = E K X N A ( Ï• ) , J ′ ∞ ( Φ ) = J N A ( Ï• ) E^('oo)(Phi)=E^(NA)(phi),quad(E^(-Ric(omega_(0))))^('oo)(Phi)=(E^(K_(X)))^(NA)(phi),quadJ^('oo)(Phi)=J^(NA)(phi)\mathbf{E}^{\prime \infty}(\Phi)=\mathbf{E}^{\mathrm{NA}}(\phi), \quad\left(\mathbf{E}^{-\operatorname{Ric}\left(\omega_{0}\right)}\right)^{\prime \infty}(\Phi)=\left(\mathbf{E}^{K_{X}}\right)^{\mathrm{NA}}(\phi), \quad \mathbf{J}^{\prime \infty}(\Phi)=\mathbf{J}^{\mathrm{NA}}(\phi)E′∞(Φ)=ENA(Ï•),(E−Ric⁡(ω0))′∞(Φ)=(EKX)NA(Ï•),J′∞(Φ)=JNA(Ï•)
Moreover, by Theorem 3.4(2), H ( Φ ) H N A ( ϕ ) H ′ ∞ ( Φ ) ≥ H N A ( Ï• ) H^('oo)(Phi) >= H^(NA)(phi)\mathbf{H}^{\prime \infty}(\Phi) \geq \mathbf{H}^{\mathrm{NA}}(\phi)H′∞(Φ)≥HNA(Ï•) so that M ( Φ ) M N A ( ϕ ) M ′ ∞ ( Φ ) ≥ M N A ( Ï• ) M^('oo)(Phi) >= M^(NA)(phi)\mathbf{M}^{\prime \infty}(\Phi) \geq \mathbf{M}^{\mathrm{NA}}(\phi)M′∞(Φ)≥MNA(Ï•).
Step 3. By Theorem 2.11, there exist models ( X m , L m ) X m , L m (X_(m),L_(m))\left(X_{m}, \mathscr{L}_{m}\right)(Xm,Lm) such that ϕ m = ϕ ( X m , L m ) Ï• m = Ï• X m , L m phi_(m)=phi_((X_(m),L_(m)))\phi_{m}=\phi_{\left(X_{m}, \mathscr{L}_{m}\right)}Ï•m=Ï•(Xm,Lm) converges to ϕ Ï• phi\phiÏ• in the strong topology and
lim m + M N A ( ϕ m ) = M N A ( ϕ ) , lim m + J N A ( ϕ m ) = J N A ( ϕ ) lim m → + ∞   M N A Ï• m = M N A ( Ï• ) , lim m → + ∞   J N A Ï• m = J N A ( Ï• ) lim_(m rarr+oo)M^(NA)(phi_(m))=M^(NA)(phi),quadlim_(m rarr+oo)J^(NA)(phi_(m))=J^(NA)(phi)\lim _{m \rightarrow+\infty} \mathbf{M}^{\mathrm{NA}}\left(\phi_{m}\right)=\mathbf{M}^{\mathrm{NA}}(\phi), \quad \lim _{m \rightarrow+\infty} \mathbf{J}^{\mathrm{NA}}\left(\phi_{m}\right)=\mathbf{J}^{\mathrm{NA}}(\phi)limm→+∞MNA(Ï•m)=MNA(Ï•),limm→+∞JNA(Ï•m)=JNA(Ï•)
Step 4. We can complete the proof by getting a contradiction to the stability assumption:
0 M ( Φ ) M N A ( ϕ ) = lim m + M N A ( ϕ m ) stability lim m + J N A ( ϕ m ) = J N A ( ϕ ) = 1 0 ≥ M ′ ∞ ( Φ ) ≥ M N A ( Ï• ) = lim m → + ∞   M N A Ï• m ≥ stability  lim m → + ∞   J N A Ï• m = J N A ( Ï• ) = 1 0 >= M^('oo)(Phi) >= M^(NA)(phi)=lim_(m rarr+oo)M^(NA)(phi_(m)) >= _("stability ")lim_(m rarr+oo)J^(NA)(phi_(m))=J^(NA)(phi)=10 \geq \mathbf{M}^{\prime \infty}(\Phi) \geq \mathbf{M}^{\mathrm{NA}}(\phi)=\lim _{m \rightarrow+\infty} \mathbf{M}^{\mathrm{NA}}\left(\phi_{m}\right) \geq_{\text {stability }} \lim _{m \rightarrow+\infty} \mathbf{J}^{\mathrm{NA}}\left(\phi_{m}\right)=\mathbf{J}^{\mathrm{NA}}(\phi)=10≥M′∞(Φ)≥MNA(Ï•)=limm→+∞MNA(Ï•m)≥stability limm→+∞JNA(Ï•m)=JNA(Ï•)=1.
There is a version of Theorem 3.7 in [45] when Aut ( X , L ) Aut ⁡ ( X , L ) Aut(X,L)\operatorname{Aut}(X, L)Aut⁡(X,L) is continuous. Moreover, it is shown in [46] that Conjecture 2.12 implies Conjecture 3.6. As mentioned earlier, if ( X , L X , L X,LX, LX,L ) is any polarized spherical manifold, Conjecture 2.12 is true and hence in this case the YTD Conjecture 3.6 is proved. Based on this fact, Delcroix [28] obtained further refined existence results in this case.
We should mention that Sean Paul (see [58]) has works that give a beautiful interpretation of the coercivity of M M M\mathbf{M}M-functional using a new notion of stability for pairs. However, it is not clear how K-stability discussed here can directly imply his stability notion.

3.3. YTD conjecture for Fano varieties

3.3.1. Non-Archimedean approach

Our proof of Theorem 3.7 is in fact modeled on a non-Archimedean approach to the uniform YTD conjecture proposed by Berman-Boucksom-Jonsson in [7]. They carried it out sucessfully for smooth Fano manifolds with discrete automorphism groups. The main advantage in the Fano case is that D N A D N A D^(NA)\mathbf{D}^{\mathrm{NA}}DNA satisfies a regularization property and can be used in place of M N A M N A M^(NA)\mathbf{M}^{\mathrm{NA}}MNA to complete the argument. Recently their work has been extended to the most general setting of log log log\loglog Fano pairs.
Theorem 3.8 ([44, 49,50]). A log Fano pair ( X , D ) ( X , D ) (X,D)(X, D)(X,D) admits a Kähler-Einstein metric if and only if it is reduced uniformly stable for all special test configurations.
Note that this combined with Theorem 2.15 also proves the K-polystable version of the YTD conjecture. Theorem 3.8 can be used to get examples of Kähler-Einstein metrics on Fano varieties with large symmetry groups (see, for example, [40]). The proof of Theorem 3.8 is much more technical than [7] because we need to overcome the difficulties caused by singularities. The first key idea is to use an approximation approach initiated in [49]. Consider the log resolution μ : X X μ : X ′ → X mu:X^(')rarr X\mu: X^{\prime} \rightarrow Xμ:X′→X as in Section 1.3 and reorganize (1.4) as
K X D ε = 1 1 + ε ( μ ( K X D ) + ε H ) =: L ε − K X ′ − D ε = 1 1 + ε μ ∗ − K X − D + ε H =: L ε -K_(X^('))-D_(epsi)=(1)/(1+epsi)(mu^(**)(-K_(X)-D)+epsi H)=:L_(epsi)-K_{X^{\prime}}-D_{\varepsilon}=\frac{1}{1+\varepsilon}\left(\mu^{*}\left(-K_{X}-D\right)+\varepsilon H\right)=: L_{\varepsilon}−KX′−Dε=11+ε(μ∗(−KX−D)+εH)=:Lε
where H = μ ( K X D ) k θ k E k H = μ ∗ − K X − D − ∑ k   θ k E k H=mu^(**)(-K_(X)-D)-sum_(k)theta_(k)E_(k)H=\mu^{*}\left(-K_{X}-D\right)-\sum_{k} \theta_{k} E_{k}H=μ∗(−KX−D)−∑kθkEk is ample by choosing appropriate { θ k } θ k {theta_(k)}\left\{\theta_{k}\right\}{θk} and D ε = D ε = D_(epsi)=D_{\varepsilon}=Dε= k ( a k + ε 1 + ε ) θ k E k ∑ k   − a k + ε 1 + ε θ k E k sum_(k)(-a_(k)+(epsi)/(1+epsi))theta_(k)E_(k)\sum_{k}\left(-a_{k}+\frac{\varepsilon}{1+\varepsilon}\right) \theta_{k} E_{k}∑k(−ak+ε1+ε)θkEk with 0 ε 1 0 ≤ ε ≪ 1 0 <= epsi≪10 \leq \varepsilon \ll 10≤ε≪1. In [49] we considered the simple case when a k a k ∈ a_(k)ina_{k} \inak∈ ( 1 , 0 ] ( − 1 , 0 ] (-1,0](-1,0](−1,0] for all k k kkk. In this case for 0 < ε 1 , ( X , D ε ) 0 < ε ≪ 1 , X ′ , D ε 0 < epsi≪1,(X^('),D_(epsi))0<\varepsilon \ll 1,\left(X^{\prime}, D_{\varepsilon}\right)0<ε≪1,(X′,Dε) is a smooth log Fano pair. A crucial calculation using the valuative criterion from Theorem 2.14 shows that (semi)stability of ( X , D ) ( X , D ) (X,D)(X, D)(X,D) implies the uniform stability of ( X , D ε ) X ′ , D ε (X^('),D_(epsi))\left(X^{\prime}, D_{\varepsilon}\right)(X′,Dε) for ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0. Moreover, we can prove a version of YTD conjecture for ( X , D ε ) X ′ , D ε (X^('),D_(epsi))\left(X^{\prime}, D_{\varepsilon}\right)(X′,Dε) and deduce that it admits a Kähler-Einstein metric. Next we take a limit as ε 0 ε → 0 epsi rarr0\varepsilon \rightarrow 0ε→0 to get a Kähler-Einstein metric on ( X , D ) ( X , D ) (X,D)(X, D)(X,D) itself. The proof of this convergence depends on technical uniform pluripotential and geometric estimates.
In [50], we dealt with the general case when D ε D ε D_(epsi)D_{\varepsilon}Dε is not necessarily effective. A key difficulty for the argument in [7] to work on singular varieties is that it is not clear how to use multiplier ideal sheaves to approximate a destabilizing geodesic ray Φ Î¦ Phi\PhiΦ when X X XXX is singular. To circumvent this difficulty, we first need to perturb Φ Î¦ Phi\PhiΦ to become a singular quasipsh potential Φ ε Φ ε Phi_(epsi)\Phi_{\varepsilon}Φε on ( X × C , p 1 L ε ) X ′ × C , p 1 ′ ∗ L ε (X^(')xxC,p_(1)^('**)L_(epsi))\left(X^{\prime} \times \mathbb{C}, p_{1}^{\prime *} L_{\varepsilon}\right)(X′×C,p1′∗Lε). Since X X ′ X^(')X^{\prime}X′ is smooth, we know how to approximate Φ ε Φ ε Phi_(epsi)\Phi_{\varepsilon}Φε by test configurations for ( X , L ε ) X ′ , L ε (X^('),L_(epsi))\left(X^{\prime}, L_{\varepsilon}\right)(X′,Lε) thanks to [7]. However, due to the ineffectiveness of D ε D ε D_(epsi)D_{\varepsilon}Dε, the remaining arguments depend more heavily on non-Archimedean analysis and some key observation on convergence of slopes. In [45] we further derived the valuative criterion for reduced uniform stability and understood how the torus action induces an action on the space of non-Archimedean potentials in order to incorporate group actions in the argument. Note that the non-Archimedean approach a priori does not prove the statement in Theorem 3.8 involving special test configurations. Fortunately, Theorem 2.13 fills this gap.
By using the fibration and approximation techniques mentioned earlier, Theorem 3.8 has been extended to the case of g g ggg-soliton on log Fano pairs in [38]. As explained in [2, 47], this can be used prove the YTD conjecture for Ricci-flat Kähler cone metrics thanks to its
equivalence to particular g g ggg-solitons (see Section 1.4). This generalizes the result of CollinsSzékelyhidi on YTD conjecture for Fano cones with isolated singularities [25].

3.3.2. Other approaches

For completeness, we briefly mention other approaches to the YTD conjecture on Fano manifolds. The classical way to solve the Kähler-Einstein equation is through various continuity methods. Traditionally, one uses Aubin's continuity method involving twisted KE metrics. A more recent continuity method uses KE metrics with edge cone singularities as proposed by Donaldson. Finally, there is a Kähler-Ricci flow approach. Tian's early works showed that the most difficult part in proving the YTD conjecture by continuity methods is to establish the algebraicity of limit objects in the Gromov-Hausdorff topology, and he had essentially reduced this difficulty to proving some partial C 0 C 0 C^(0)C^{0}C0-estimates. The partial C 0 C 0 C^(0)C^{0}C0 estimates were later proved in different settings, starting with Donaldson-Sun's work in the Kähler-Einstein case, which led to the solution of the YTD conjecture for smooth Fano manifolds in [24,65]. Moreover, the partial C 0 C 0 C^(0)C^{0}C0-estimates has applications in constructing moduli spaces of smoothable Kähler-Einstein varieties and proving quasiprojectivity of the moduli spaces of KE manifolds, and these applications preclude the algebraic approach mentioned earlier (see [68]). We also refer to [31,66] for surveys on related topics in this approach.
Very recently, yet another quantization approach is carried out by Kewei Zhang based partly on an earlier work of Rubinstein-Tian-Zhang. Zhang considered an analytic invariant of Moser-Trudinger type, namely
δ A ( X ) = sup { c : sup φ H X e c ( φ E ( φ ) ) < + } δ A ( X ) = sup c : sup φ ∈ H   ∫ X   e − c ( φ − E ( φ ) ) < + ∞ delta^(A)(X)=s u p{c:s u p_(varphi inH)int_(X)e^(-c(varphi-E(varphi))) < +oo}\delta^{A}(X)=\sup \left\{c: \sup _{\varphi \in \mathscr{H}} \int_{X} e^{-c(\varphi-\mathbf{E}(\varphi))}<+\infty\right\}δA(X)=sup{c:supφ∈H∫Xe−c(φ−E(φ))<+∞}
It is easy to show that the coercivity of D D D\mathbf{D}D-functional is equivalent to δ A ( X ) > 1 δ A ( X ) > 1 delta^(A)(X) > 1\delta^{A}(X)>1δA(X)>1. The authors of [61] introduced a quantization δ m A ( X ) δ m A ( X ) delta_(m)^(A)(X)\delta_{m}^{A}(X)δmA(X) by using a quantization of E E E\mathbf{E}E on the space of Bergman metrics, and further proved δ m A ( X ) = δ m ( X ) δ m A ( X ) = δ m ( X ) delta_(m)^(A)(X)=delta_(m)(X)\delta_{m}^{A}(X)=\delta_{m}(X)δmA(X)=δm(X). Using some deep results in complex geometry including Tian's work on Bergman kernels and Berndtsson's subharmonicity theorem, it is proved in [73] that lim m + δ m A ( X ) = δ A ( X ) lim m → + ∞   δ m A ( X ) = δ A ( X ) lim_(m rarr+oo)delta_(m)^(A)(X)=delta^(A)(X)\lim _{m \rightarrow+\infty} \delta_{m}^{A}(X)=\delta^{A}(X)limm→+∞δmA(X)=δA(X). Combining these discussions with the algebraic convergence result of Blum-Jonsson and the valuative criterion of uniform stability of Fujita discussed earlier, Zhang gets δ A ( X ) = δ ( X ) δ A ( X ) = δ ( X ) delta^(A)(X)=delta(X)\delta^{A}(X)=\delta(X)δA(X)=δ(X) and completes the proof of uniform version of YTD conjecture for smooth Fano manifolds. It would be interesting to extend this approach to the more general case (i.e., Fano varieties with continuous automorphism groups).
We finish by remarking that it is of interest to apply the ideas and methods from the above two approaches to study the YTD conjecture for general polarizations. For the approach involving partial C 0 C 0 C^(0)C^{0}C0-estimates, the geometry is complicated by collapsing phenomenon in the Gromov-Hausdorff convergence with only scalar curvature bounds, which is very difficult to study with current techniques. For the quantization approach, there were some attempts by Mabuchi in several works. But the precise picture seems again unclear.

ACKNOWLEDGMENTS

The author thanks Gang Tian, Chenyang Xu, Feng Wang, Xiaowei Wang, Jiyuan Han, and Yuchen Liu for collaborations on works discussed in this survey. He also thanks many mathematicians for various discussions related to our works: Vestislav Apostolov, Robert Berman, Sébastien Boucksom, Tamás Darvas, Simon Donaldson, Kento Fujita, HansJoachim Hein, Mattias Jonsson, László Lempert, Mircea Mustaţă, Yuji Odaka, Yanir Rubinstein, Jian Song, Song Sun, Jun Yu, and many others. A large part of the author's works discussed here were carried out at the Department of Mathematics, Purdue University, and he is grateful to his colleagues and friends there for creating a great environment of research.

FUNDING

This work is partially supported by NSF grant (DMS-181086) and an Alfred Sloan Fellowship.

REFERENCES

[1] H. Ahmadinezhad and Z. Zhuang, K-stability of Fano varieties via admissible flags. 2020, arXiv:2003.13788.
[2] V. Apostolov, S. Jubert, and A. Lahdili, Weighted K-stability and coercivity with applications to extremal Kähler and Sasaki metrics. 2021, arXiv:2104.09709.
[3] C. Araujo, A.-M. Castravet, I. Cheltsov, K. Fujita, A.-S. Kaloghiros, J. MartinezGarcia, C. Shramov, H. Süss, and N. Viswanathan, The Calabi problem for Fano threefolds. 2021, MPIM preprint.
[4] R. Berman, K-polystability of Q Q Q\mathbb{Q}Q-Fano varieties admitting Kähler-Einstein metrics. Invent. Math. 203 (2015), no. 3, 973-1025.
[5] R. Berman and R. Berndtsson, Convexity of the K-energy on the space of Kähler metrics. J. Amer. Math. Soc. 30 (2017), 1165-1196.
[6] R. Berman, S. Boucksom, P. Eyssidieux, V. Guedj, and A. Zeriahi, KählerEinstein metrics and the Kähler-Ricci flow on log Fano varieties. J. Reine Angew. Math. 751 (2019), 27-89.
[7] R. Berman, S. Boucksom, and M. Jonsson, A variational approach to the YauTian-Donaldson conjecture. J. Amer. Math. Soc. 34 (2020), no. 3, 605-652.
[8] R. Berman, T. Darvas, and C. H. Lu, Convexity of the extended K-energy and the large time behaviour of the weak Calabi flow. Geom. Topol. 21 (2017), 2945-2988.
[9] R. Berman, T. Darvas, and C. H. Lu, Regularity of weak minimizers of the Kenergy and applications to properness and K-stability. Ann. Sci. Éc. Norm. Supér. (4) 53 (2020), no. 2, 267-289.
[10] R. Berman and D. Witt Nyström, Complex optimal transport and the pluripotential theory of Kähler-Ricci solitons, Preprint, 2014, arXiv:1401.8264.
[11] H. Blum, Existence of valuations with smallest normalized volume. Compos. Math. 154 (2018), no. 4, 820-849.
[12] H. Blum and M. Jonsson, Thresholds, valuations, and K-stability. Adv. Math. 365 (2020), 107062.
[13] H. Blum, Y. Liu, and C. Xu, Openness of K-semistability for Fano varieties. 2019, arXiv:1907.02408.
[14] H. Blum, Y. Liu, C. Xu, and Z. Zhuang, The existence of the Kähler-Ricci soliton degeneration. 2021, arXiv:2103.15278.
[15] H. Blum and C. Xu, Uniqueness of K-polystable degenerations of Fano varieties. Ann. of Math. (2) 190 (2019), no. 2, 609-656.
[16] S. Boucksom, Variational and non-Archimedean aspects of the Yau-Tian-Donaldson conjecture. In Proceedings of the International Congress of Mathematicians - Rio de Janeiro 2018. Vol. II. Invited lectures, pp. 591-617, World Sci. Publ., Hackensack, NJ, 2018.
[17] S. Boucksom, C. Favre, and M. Jonsson, Differentiability of volumes of divisors and a problem of Teissier. J. Algebraic Geom. 18 (2009), no. 2, 279-308.
[18] S. Boucksom, C. Favre, and M. Jonsson, Solution to a non-Archimedean MongeAmpère equation. J. Amer. Math. Soc. 28 (2015), 617-667.
[19] S. Boucksom, T. Hisamoto, and M. Jonsson, Uniform K-stability, DuistermaatHeckman measures and singularities of pairs. Ann. Inst. Fourier (Grenoble) 67 (2017), 87-139.
[20] S. Boucksom, T. Hisamoto, and M. Jonsson, Uniform K-stability and asymptotics of energy functionals in Kähler geometry. J. Eur. Math. Soc. (JEMS) 21 (2019), no. 9, 2905-2944.
[21] S. Boucksom and M. Jonsson, A non-Archimedean approach to K-stability. 2018, arXiv:1805.11160v1.
[22] S. Boucksom and M. Jonsson, Global pluripotential theory over a trivially valued field. 2021, arXiv:1801.08229v2.
[23] X. Chen and J. Cheng, On the constant scalar curvature Kähler metrics. J. Amer. Math. Soc. 34 (2021), no. 4, 909-1009.
[24] X. Chen, S. K. Donaldson, and S. Sun, Kähler-Einstein metrics on Fano manifolds, I-III. J. Amer. Math. Soc. 28 (2015), 183-197, 199-234, 235-278.
[25] T. Collins and G. Székelyhidi, Sasaki-Einstein metrics and K-stability. Geom. Topol. 23 (2019), no. 3, 1339-1413.
[26] T. Darvas, The Mabuchi geometry of finite energy classes. Adv. Math. 285 (2015), 182 219 182 − 219 182-219182-219182−219.
[27] T. Darvas and Y. Rubinstein, Tian's properness conjectures and Finsler geometry of the space of Kähler metrics. J. Amer. Math. Soc. 30 (2017), 347-387.
[28] T. Delcroix, Uniform K-stability of polarized spherical varieties. 2020, arXiv:2009.06463.
[29] R. Dervan, Uniform stability of twisted constant scalar curvature Kähler metrics. Int. Math. Res. Not. 2016 (2016), no. 15, 4728-4783.
[30] S. Donaldson, Scalar curvature and stability of toric varieties. J. Differential Geom. 62 (2002), no. 2, 289-349.
[31] S. Donaldson, Kähler-Einstein metrics and algebraic geometry. In Current developments in mathematics 2015, pp. 1-25, International Press, Somerville, MA, 2016.
[32] P. Eyssidieux, V. Guedj, and A. Zeriahi, Singular Kähler-Einstein metrics. J. Amer. Math. Soc. 22 (2009), 607-639.
[33] K. Fujita, Optimal bounds for the volumes of Kähler-Einstein Fano manifolds. Amer. J. Math. 140 (2018), no. 2, 391-414.
[34] K. Fujita, Uniform K-stability and plt blowups of log Fano pairs. Kyoto J. Math. 59 (2019), no. 2, 399-418.
[35] K. Fujita, A valuative criterion for uniform K-stability of Q Q Q\mathbb{Q}Q-Fano varieties. J. Reine Angew. Math. 751 (2019), 309-338.
[36] K. Fujita and Y. Odaka, On the K-stability of Fano varieties and anticanonical divisors. Tohoku Math. J. 70 (2018), no. 4, 511-521.
[37] J. Han and C. Li, Algebraic uniqueness of Kähler-Ricci flow limits and optimal degenerations of Fano varieties. 2009, arXiv:2009.01010.
[38] J. Han and C. Li, On the Yau-Tian-Donaldson conjecture for g g ggg-weighted KählerRicci soliton equations. Comm. Pure Appl. Math. (to appear), arXiv:2006.00903.
[39] T. Hisamoto, Stability and coercivity for toric polarizations. 2016, arXiv:1610.07998.
[40] N. Ilten and H. Süß, K-stability for Fano manifolds with torus action of complexity 1. Duke Math. J. 166 (2017), no. 1, 177-204.
[41] M. Jonsson and M. Mustaţă, Valuations and asymptotic invariants for sequences of ideals. Ann. Inst. Fourier 62 (2012), no. 6, 2145-2209.
[42] C. Li, K-semistability is equivariant volume minimization. Duke Math. J. 166 (2017), no. 16, 3147-3218.
[43] C. Li, Minimizing normalized volume of valuations. Math. Z. 289 (2018), no. 1-2, 491-513.
[44] C. Li, G-uniform stability and Kähler-Einstein metrics on Fano varieties. Invent. Math. (to appear), arXiv:1907.09399v3.
[45] C. Li, Geodesic rays and stability in the cscK problem. Ann. Sci. Éc. Norm. Supér. (to appear), arXiv:2001.01366.
[46] C. Li, K-stability and Fujita approximation. 2021, arXiv:2102.09457.
[47] C. Li, Notes on weighted Kähler-Ricci solitons and application to Ricci-flat Kähler cone metrics. 2021, arXiv:2107.02088.
[48] C. Li, Y. Liu, and C. Xu, A guided tour to normalized volume. In Geometric analysis, pp. 167-219, Progr. Math. 333, Springer, Cham, 2020.
[49] C. Li, G. Tian, and F. Wang, On the Yau-Tian-Donaldson conjecture for singular Fano varieties. Comm. Pure Appl. Math. 74 (2020), no. 8, 1748-1800.
[50] C. Li, G. Tian, and F. Wang, The uniform version of Yau-Tian-Donaldson conjecture for singular Fano varieties. Peking Math. J. (to appear), arXiv:1903.01215.
[51] C. Li, X. Wang, and C. Xu, Algebraicity of metric tangent cones and equivariant K-stability. J. Amer. Math. Soc. 34 (2021), no. 4, 1175-1214.
[52] C. Li and C. Xu, Special test configurations and K-stability of Fano varieties. Ann. of Math. (2) 180 (2014), no. 1, 197-232.
[53] C. Li and C. Xu, Stability of valuations: Higher rational rank. Peking Math. J. 1 (2018), no. 1, 1-79.
[54] C. Li and C. Xu, Stability of valuations and Kollár components. J. Eur. Math. Soc. (JEMS) 22 (2020), no. 8, 2573-2627.
[55] Y. Liu, The volume of singular Kähler-Einstein Fano varieties. Compos. Math. 154 (2018), no. 6, 1131-1158.
[56] Y. Liu, C. Xu, and Z. Zhuang, Finite generation for valuations computing stability thresholds and applications to K-stability. 2021, arXiv:2012.09405.
[57] D. Martelli, J. Sparks, and S.-T. Yau, Sasaki-Einstein manifolds and volume minimisation. Comm. Math. Phys. 280 (2008), no. 3, 611-673.
[58] S. T. Paul, Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics. Ann. of Math. 175 (2012), no. 1, 255-296.
[59] D. H. Phong, J. Ross, and J. Sturm, Deligne pairings and the Knudsen-Mumford expansion. J. Differential Geom. 78 (2008), 475-496.
[60] D. H. Phong and J. Sturm, Test configurations for K-stability and geodesic rays. J. Symplectic Geom. 5 (2007), no. 2, 221-247.
[61] Y. A. Rubinstein, G. Tian, and K. Zhang, Basis divisors and balanced metrics. J. Reine Angew. Math., 778 (2021), 171-218.
[62] S. Sun, Degenerations and moduli spaces in Kähler geometry. In Proceedings of the International Congress of Mathematicians - Rio de Janeiro 2018, Vol. II. Invited lectures, pp. 993-1012, World Sci. Publ., Hackensack, NJ, 2018.
[63] G. Székelyhidi, Filtrations and test configurations, with an appendix by S. Boucksom. Math. Ann. 362 (2015), no. 1-2, 451-484.
[64] G. Tian, Kähler-Einstein metrics with positive scalar curvature. Invent. Math. 130 (1997), 239-265.
[65] G. Tian, K-stability and Kähler-Einstein metrics. Comm. Pure Appl. Math. 68 (2015), no. 7, 1085-1156.
[66] G. Tian, Kähler-Einstein metrics on Fano manifolds. Jpn. J. Math. 10 (2015), no. 1, 1-41.
[67] G. Tian, K-stability implies CM-stability. In Geometry, analysis and probability, pp. 245-261, Progr. Math. 310, Springer, Cham, 2017.
[68] X. Wang, GIT stability, K-stability and the moduli space of Fano varieties. In Moduli of K K KKK-stable varieties, pp. 153-181, Springer INdAM Series 31, Springer, Cham, 2019.
[69] D. Witt Nyström, Test configuration and Okounkov bodies. Compos. Math. 148 (2012), no. 6, 1736-1756.
[70] C. Xu, K-stability of Fano varieties: an algebro-geometric approach. EMS Surv. Math. Sci. 8 (2021), no. 1-2, 265-354.
[71] C. Xu, A minimizing valuation is quasi-monomial. Ann. of Math. 191 (2020), no. 3, 1003-1030.
[72] C. Xu and Z. Zhuang, Uniqueness of the minimizer of the normalized volume function. Camb. J. Math. 9 (2021), no. 1, 149-176.
[73] K. Zhang, A quantization proof of the uniform Yau-Tian-Donaldson conjecture. 2021, arXiv:2102.02438v2.
[74] X. Zhu, Kähler-Ricci flow on Fano manifolds. In Proceedings of the International Congress of Mathematicians 2022 (to appear), 2022.

CHI LI

Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd. Piscataway, NJ 08854-8019, USA, chi.li @ rutgers.edu

THE DOUBLE RAMIFICATION CYCLE FORMULA

AARON PIXTON

ABSTRACT

The double ramification cycle D R g ( A ) = D R g ( μ , ν ) D R g ( A ) = D R g ( μ , ν ) DR_(g)(A)=DR_(g)(mu,nu)\mathrm{DR}_{g}(A)=\mathrm{DR}_{g}(\mu, \nu)DRg(A)=DRg(μ,ν) is a cycle in the moduli space of stable curves parametrizing genus g g ggg curves admitting a map to P 1 P 1 P^(1)\mathbb{P}^{1}P1 with specified ramification profiles μ , ν μ , ν mu,nu\mu, \nuμ,ν over two points. In 2016, Janda, Pandharipande, Zvonkine, and the author proved a formula expressing the double ramification cycle in terms of basic tautological classes, answering a question of Eliashberg from 2001. This formula has an intricate combinatorial shape involving an unusual way to sum divergent series using polynomial interpolation. Here we give some motivation for where this formula came from, relating it both to an older partial formula of Hain and to Givental's R-matrix action on cohomological field theories.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 14N35; Secondary 14H10, 14C17

KEYWORDS

Moduli of curves, double ramification cycle

1. INTRODUCTION

Let g , n g , n g,ng, ng,n be nonnegative integers satisfying 2 g 2 + n > 0 2 g − 2 + n > 0 2g-2+n > 02 g-2+n>02g−2+n>0, so that the moduli space M ¯ g , n M ¯ g , n bar(M)_(g,n)\overline{\mathcal{M}}_{g, n}M¯g,n of stable curves of genus g g ggg with n n nnn markings is nonempty. Let A = ( a 1 , , a n ) Z n A = a 1 , … , a n ∈ Z n A=(a_(1),dots,a_(n))inZ^(n)A=\left(a_{1}, \ldots, a_{n}\right) \in \mathbb{Z}^{n}A=(a1,…,an)∈Zn be a vector of n n nnn integers satisfying a 1 + + a n = 0 a 1 + ⋯ + a n = 0 a_(1)+cdots+a_(n)=0a_{1}+\cdots+a_{n}=0a1+⋯+an=0. In this paper we will be interested in a Chow cycle class
DR g ( A ) A g ( M ¯ g , n ) DR g ⁡ ( A ) ∈ A g M ¯ g , n DR_(g)(A)inA^(g)( bar(M)_(g,n))\operatorname{DR}_{g}(A) \in A^{g}\left(\overline{\mathcal{M}}_{g, n}\right)DRg⁡(A)∈Ag(M¯g,n)
that depends on this data.
There are two main perspectives on how to think about and define DR g ( A ) DR g ⁡ ( A ) DR_(g)(A)\operatorname{DR}_{g}(A)DRg⁡(A), the double ramification cycle. The first is the source of its name; we think of it as parametrizing the genus g g ggg curves C C CCC that admit a finite map C P 1 C → P 1 C rarrP^(1)C \rightarrow \mathbb{P}^{1}C→P1 with specified ramification profiles μ , ν μ , ν mu,nu\mu, \nuμ,ν over two points (say 0 and ∞ oo\infty∞ ). These two ramification profiles are encoded in the vector A A AAA : we can take the positive and negative components of A A AAA separately to get two partitions of equal size. The marked points with nonzero a i a i a_(i)a_{i}ai should then be the points in C C CCC lying above 0 and ∞ oo\infty∞, while the marked points with a i = 0 a i = 0 a_(i)=0a_{i}=0ai=0 are unconstrained. Ramification above points other than 0 and ∞ oo\infty∞ is unconstrained.
The above description defines a double ramification locus inside the moduli space of smooth curves M g , n M g , n M_(g,n)\mathcal{M}_{g, n}Mg,n that is usually (but not always) of pure codimension g g ggg. To extend this to a codimension g g ggg class on M ¯ g , n M ¯ g , n bar(M)_(g,n)\overline{\mathcal{M}}_{g, n}M¯g,n, we can use the virtual class in relative GromovWitten theory. There is a moduli space of stable (rubber) maps to P 1 P 1 P^(1)\mathbb{P}^{1}P1 with given marked ramification over two points, M ¯ g , n ( P 1 / { 0 , } , μ , ν ) M ¯ g , n P 1 / { 0 , ∞ } , μ , ν ∼ bar(M)_(g,n)(P^(1)//{0,oo},mu,nu)^(∼)\overline{\mathcal{M}}_{g, n}\left(\mathbb{P}^{1} /\{0, \infty\}, \mu, \nu\right)^{\sim}M¯g,n(P1/{0,∞},μ,ν)∼, equipped with a forgetful map p p ppp : M ¯ g , n ( P 1 / { 0 , } , μ , ν ) M ¯ g , n M ¯ g , n P 1 / { 0 , ∞ } , μ , ν ∼ → M ¯ g , n bar(M)_(g,n)(P^(1)//{0,oo},mu,nu)^(∼)rarr bar(M)_(g,n)\overline{\mathcal{M}}_{g, n}\left(\mathbb{P}^{1} /\{0, \infty\}, \mu, \nu\right)^{\sim} \rightarrow \overline{\mathcal{M}}_{g, n}M¯g,n(P1/{0,∞},μ,ν)∼→M¯g,n, and the double ramification cycle can be taken to be the pushforward under this map of the virtual class,
DR g ( A ) = p [ M ¯ g , n ( P 1 / { 0 , } , μ , v ) ] v i r A g ( M ¯ g , n ) DR g ⁡ ( A ) = p ∗ M ¯ g , n P 1 / { 0 , ∞ } , μ , v ∼ v i r ∈ A g M ¯ g , n DR_(g)(A)=p_(**)[ bar(M)_(g,n)(P^(1)//{0,oo},mu,v)^(∼)]^(vir)inA^(g)( bar(M)_(g,n))\operatorname{DR}_{g}(A)=p_{*}\left[\overline{\mathcal{M}}_{g, n}\left(\mathbb{P}^{1} /\{0, \infty\}, \mu, v\right)^{\sim}\right]^{\mathrm{vir}} \in A^{g}\left(\overline{\mathcal{M}}_{g, n}\right)DRg⁡(A)=p∗[M¯g,n(P1/{0,∞},μ,v)∼]vir∈Ag(M¯g,n)
The second perspective on DR g ( A ) DR g ⁡ ( A ) DR_(g)(A)\operatorname{DR}_{g}(A)DRg⁡(A) is via Abel-Jacobi maps. Let X g A g X g → A g X_(g)rarrA_(g)\mathcal{X}_{g} \rightarrow \mathcal{A}_{g}Xg→Ag be the universal abelian variety of dimension g g ggg. Then the data in the vector A A AAA can be used to define a morphism j A : M g , n X g j A : M g , n → X g j_(A):M_(g,n)rarrX_(g)j_{A}: \mathcal{M}_{g, n} \rightarrow \mathcal{X}_{g}jA:Mg,n→Xg by
( C , p 1 , , p n ) ( Jac ( C ) , O C ( a 1 p 1 + + a n p n ) ) C , p 1 , … , p n ↦ Jac ⁡ ( C ) , O C a 1 p 1 + ⋯ + a n p n (C,p_(1),dots,p_(n))|->(Jac(C),O_(C)(a_(1)p_(1)+cdots+a_(n)p_(n)))\left(C, p_{1}, \ldots, p_{n}\right) \mapsto\left(\operatorname{Jac}(C), \mathcal{O}_{C}\left(a_{1} p_{1}+\cdots+a_{n} p_{n}\right)\right)(C,p1,…,pn)↦(Jac⁡(C),OC(a1p1+⋯+anpn))
The double ramification locus is then the inverse image under this map of the zero section Z g Z g Z_(g)Z_{g}Zg of X g X g X_(g)\mathcal{X}_{g}Xg, since C C CCC admits a map to P 1 P 1 P^(1)\mathbb{P}^{1}P1 with the given ramification profiles if and only if O C ( a 1 p 1 + + a n p n ) O C a 1 p 1 + ⋯ + a n p n O_(C)(a_(1)p_(1)+cdots+a_(n)p_(n))\mathcal{O}_{C}\left(a_{1} p_{1}+\cdots+a_{n} p_{n}\right)OC(a1p1+⋯+anpn) is trivial.
This Abel-Jacobi map extends easily to M g , n c t M g , n c t M_(g,n)^(ct)\mathcal{M}_{g, n}^{\mathrm{ct}}Mg,nct, the moduli space of curves of compact type (those with compact Jacobians), but using this perspective to define the double ramification cycle on all of M ¯ g , n M ¯ g , n bar(M)_(g,n)\overline{\mathcal{M}}_{g, n}M¯g,n requires more work. It also is not obvious that constructing DR g ( A ) DR g ⁡ ( A ) DR_(g)(A)\operatorname{DR}_{g}(A)DRg⁡(A) in this way will give the same class as that given by relative Gromov-Witten theory, even after restriction to M g , n c t M g , n c t M_(g,n)^(ct)\mathcal{M}_{g, n}^{\mathrm{ct}}Mg,nct. For one approach to these questions using logarithmic and tropical geometry, see the work of Marcus and Wise [13].
Eliashberg proposed the problem of giving a formula for the double ramification cycle in 2001, in the context of symplectic field theory. This problem was solved by Janda, Pandharipande, Zvonkine, and the author in 2016 [11], giving an explicit combinatorial formula for the double ramification cycle. This formula has an unexpected form-an additional
integer parameter r > 0 r > 0 r > 0r>0r>0 is introduced, then an expression is written down that becomes polynomial in r r rrr for r r rrr sufficiently large, and finally this polynomial is specialized to r = 0 r = 0 r=0r=0r=0. Subsequent papers extending or generalizing the double ramification cycle formula in various ways (e.g., [ 2 , 5 , 12 ] [ 2 , 5 , 12 ] [2,5,12][2,5,12][2,5,12] ) have left the basic combinatorial structure of the formula virtually unchanged. The purpose of this paper is to discuss this structure and give some motivation for where it comes from.
In Section 2, we review the tautological classes in the Chow ring of the moduli space of stable curves. In Section 3, we discuss results leading up to the formula of [11], most notably Hain's formula for the compact type double ramification cycle. Section 4 is the heart of the paper and consists of an extended discussion motivating the shape of the double ramification cycle formula. We conclude in Section 5 by stating the formula and briefly explaining how its proof in [11] is related to some of the motivation in Section 4.

2. TAUTOLOGICAL CLASSES

2.1. Preliminaries

In this section we review the language in which the double ramification cycle formula is written. This is the language of the tautological ring, a subring R ( M ¯ g , n ) A ( M ¯ g , n ) R ∗ M ¯ g , n ⊆ A ∗ M ¯ g , n R^(**)( bar(M)_(g,n))subeA^(**)( bar(M)_(g,n))R^{*}\left(\overline{\mathcal{M}}_{g, n}\right) \subseteq A^{*}\left(\overline{\mathcal{M}}_{g, n}\right)R∗(M¯g,n)⊆A∗(M¯g,n) containing most classes that arise naturally in geometry.
Following Faber and Pandharipande [6], the tautological rings R ( M ¯ g , n ) R ∗ M ¯ g , n R^(**)( bar(M)_(g,n))R^{*}\left(\overline{\mathcal{M}}_{g, n}\right)R∗(M¯g,n) can be defined simultaneously for all g , n 0 g , n ≥ 0 g,n >= 0g, n \geq 0g,n≥0 satisfying 2 g 2 + n > 0 2 g − 2 + n > 0 2g-2+n > 02 g-2+n>02g−2+n>0 as the smallest subrings of the Chow rings A ( M ¯ g , n ) A ∗ M ¯ g , n A^(**)( bar(M)_(g,n))A^{*}\left(\overline{\mathcal{M}}_{g, n}\right)A∗(M¯g,n) closed under pushforward by forgetful maps M ¯ g , n + 1 M ¯ g , n M ¯ g , n + 1 → M ¯ g , n bar(M)_(g,n+1)rarr bar(M)_(g,n)\overline{\mathcal{M}}_{g, n+1} \rightarrow \overline{\mathcal{M}}_{g, n}M¯g,n+1→M¯g,n
and gluing maps M ¯ g , n + 2 M ¯ g + 1 , n M ¯ g , n + 2 → M ¯ g + 1 , n bar(M)_(g,n+2)rarr bar(M)_(g+1,n)\overline{\mathcal{M}}_{g, n+2} \rightarrow \overline{\mathcal{M}}_{g+1, n}M¯g,n+2→M¯g+1,n or M ¯ g 1 , n 1 + 1 × M ¯ g 2 , n 2 + 1 M ¯ g 1 + g 2 , n 1 + n 2 M ¯ g 1 , n 1 + 1 × M ¯ g 2 , n 2 + 1 → M ¯ g 1 + g 2 , n 1 + n 2 bar(M)_(g_(1),n_(1)+1)xx bar(M)_(g_(2),n_(2)+1)rarr bar(M)_(g_(1)+g_(2),n_(1)+n_(2))\overline{\mathcal{M}}_{g_{1}, n_{1}+1} \times \overline{\mathcal{M}}_{g_{2}, n_{2}+1} \rightarrow \overline{\mathcal{M}}_{g_{1}+g_{2}, n_{1}+n_{2}}M¯g1,n1+1×M¯g2,n2+1→M¯g1+g2,n1+n2. Our discussions of tautological classes will use a more explicit description of them. Graber and Pandharipande [8, APPENDIX A] gave a set of additive generators and a multiplication law satisfied by these generators.
These additive generators are formed from three ingredients: psi classes, kappa classes, and generalized gluing maps corresponding to stable graphs. The psi classes ψ i A 1 ( M ¯ g , n ) , i = 1 , , n ψ i ∈ A 1 M ¯ g , n , i = 1 , … , n psi_(i)inA^(1)( bar(M)_(g,n)),i=1,dots,n\psi_{i} \in A^{1}\left(\overline{\mathcal{M}}_{g, n}\right), i=1, \ldots, nψi∈A1(M¯g,n),i=1,…,n correspond to the n n nnn marked points and are defined as the first Chern classes of the cotangent line bundles to the curves at those points. The ArbarelloCornalba [1] kappa classes are then the pushforwards of powers of psi classes,
κ a := π ( ψ n + 1 a + 1 ) A a ( M ¯ g , n ) κ a := Ï€ ∗ ψ n + 1 a + 1 ∈ A a M ¯ g , n kappa_(a):=pi_(**)(psi_(n+1)^(a+1))inA^(a)( bar(M)_(g,n))\kappa_{a}:=\pi_{*}\left(\psi_{n+1}^{a+1}\right) \in A^{a}\left(\overline{\mathcal{M}}_{g, n}\right)κa:=π∗(ψn+1a+1)∈Aa(M¯g,n)
where π : M ¯ g , n + 1 M ¯ g , n Ï€ : M ¯ g , n + 1 → M ¯ g , n pi: bar(M)_(g,n+1)rarr bar(M)_(g,n)\pi: \overline{\mathcal{M}}_{g, n+1} \rightarrow \overline{\mathcal{M}}_{g, n}Ï€:M¯g,n+1→M¯g,n forgets the last marking. The kappa classes will not appear in any of the formulas in this paper.
The tautological ring of the moduli space of smooth marked curves, R ( M g , n ) R ∗ M g , n R^(**)(M_(g,n))R^{*}\left(\mathcal{M}_{g, n}\right)R∗(Mg,n), is the ring generated by these ψ i ψ i psi_(i)\psi_{i}ψi and κ a κ a kappa_(a)\kappa_{a}κa. To extend this to R ( M ¯ g , n ) R ∗ M ¯ g , n R^(**)( bar(M)_(g,n))R^{*}\left(\overline{\mathcal{M}}_{g, n}\right)R∗(M¯g,n) we need classes supported on boundary strata.

2.2. Stable graphs

A stable graph Γ Î“ Gamma\GammaΓ is the combinatorial data of a boundary stratum in M ¯ g , n M ¯ g , n bar(M)_(g,n)\overline{\mathcal{M}}_{g, n}M¯g,n. It consists of the following:
(1) a set of vertices V ( Γ ) V ( Γ ) V(Gamma)V(\Gamma)V(Γ);
(2) a genus g v 0 g v ≥ 0 g_(v) >= 0g_{v} \geq 0gv≥0 at each vertex v V ( Γ ) v ∈ V ( Γ ) v in V(Gamma)v \in V(\Gamma)v∈V(Γ);
(3) a set of half-edges H ( Γ ) H ( Γ ) H(Gamma)H(\Gamma)H(Γ);
(4) an incidence map H ( Γ ) V ( Γ ) H ( Γ ) → V ( Γ ) H(Gamma)rarr V(Gamma)H(\Gamma) \rightarrow V(\Gamma)H(Γ)→V(Γ);
(5) a partition of H ( Γ ) H ( Γ ) H(Gamma)H(\Gamma)H(Γ) into sets of size 2 (called edges, the set of which is denoted E ( Γ ) ) E ( Γ ) ) E(Gamma))E(\Gamma))E(Γ)) and sets of size 1 (called legs);
(6) a bijection between the set of legs and { 1 , , n } { 1 , … , n } {1,dots,n}\{1, \ldots, n\}{1,…,n}.
The underlying graph is required to be connected. The stability constraint is that
2 g v 2 + n v > 0 2 g v − 2 + n v > 0 2g_(v)-2+n_(v) > 02 g_{v}-2+n_{v}>02gv−2+nv>0
at each vertex v v vvv, where n v n v n_(v)n_{v}nv is the number of half-edges incident to v v vvv. The genera are constrained by the identity
2 g 2 + n = v V ( Γ ) ( 2 g v 2 + n v ) 2 g − 2 + n = ∑ v ∈ V ( Γ )   2 g v − 2 + n v 2g-2+n=sum_(v in V(Gamma))(2g_(v)-2+n_(v))2 g-2+n=\sum_{v \in V(\Gamma)}\left(2 g_{v}-2+n_{v}\right)2g−2+n=∑v∈V(Γ)(2gv−2+nv)
or equivalently that g v g v = h 1 ( Γ ) g − ∑ v   g v = h 1 ( Γ ) g-sum_(v)g_(v)=h^(1)(Gamma)g-\sum_{v} g_{v}=h^{1}(\Gamma)g−∑vgv=h1(Γ), the first Betti number of the graph. Such a stable graph Γ Î“ Gamma\GammaΓ corresponds to a generalized gluing map
ξ Γ : v V ( Γ ) M ¯ g v , n v M ¯ g , n ξ Γ : ∏ v ∈ V ( Γ )   M ¯ g v , n v → M ¯ g , n xi_(Gamma):prod_(v in V(Gamma)) bar(M)_(g_(v),n_(v))rarr bar(M)_(g,n)\xi_{\Gamma}: \prod_{v \in V(\Gamma)} \overline{\mathcal{M}}_{g_{v}, n_{v}} \rightarrow \overline{\mathcal{M}}_{g, n}ξΓ:∏v∈V(Γ)M¯gv,nv→M¯g,n
We can then consider classes
ξ Γ ( α ) A ( M ¯ g , n ) ξ Γ ∗ ( α ) ∈ A ∗ M ¯ g , n xi_(Gamma**)(alpha)inA^(**)( bar(M)_(g,n))\xi_{\Gamma *}(\alpha) \in A^{*}\left(\overline{\mathcal{M}}_{g, n}\right)ξΓ∗(α)∈A∗(M¯g,n)
where Γ Î“ Gamma\GammaΓ is a stable graph and α α alpha\alphaα is a monomial in the psi and kappa classes on the M ¯ g v , n v M ¯ g v , n v bar(M)_(g_(v),n_(v))\overline{\mathcal{M}}_{g_{v}, n_{v}}M¯gv,nv factors. These are the additive generators for the tautological ring considered in [8].

2.3. Compact type

The moduli space of curves of compact type, denoted M g , n c t M g , n c t M_(g,n)^(ct)\mathcal{M}_{g, n}^{\mathrm{ct}}Mg,nct, is the open subscheme of M ¯ g , n M ¯ g , n bar(M)_(g,n)\overline{\mathcal{M}}_{g, n}M¯g,n consisting of those curves whose dual graph is a tree. Its tautological ring R ( M g , n c t ) R ∗ M g , n c t R^(**)(M_(g,n)^(ct))R^{*}\left(\mathcal{M}_{g, n}^{\mathrm{ct}}\right)R∗(Mg,nct) is the image of R ( M ¯ g , n ) R ∗ M ¯ g , n R^(**)( bar(M)_(g,n))R^{*}\left(\overline{\mathcal{M}}_{g, n}\right)R∗(M¯g,n) under restriction, so it is additively generated by classes ξ Γ ( α ) ξ Γ ∗ ( α ) xi_(Gamma_(**))(alpha)\xi_{\Gamma_{*}}(\alpha)ξΓ∗(α) as above where Γ Î“ Gamma\GammaΓ is a tree.
It will be convenient for us to have notation for the compact type boundary divisor classes when stating Hain's formula below, (3.2). If Γ Î“ Gamma\GammaΓ is a stable graph with 2 vertices and 1 edge and one of the vertices is genus h h hhh and has those legs with markings in a set S 1 , 2 , , n S ⊆ 1 , 2 , … , n S sube1,2,dots,nS \subseteq 1,2, \ldots, nS⊆1,2,…,n, let δ h , S = ξ Γ ( 1 ) δ h , S = ξ Γ ∗ ( 1 ) delta_(h,S)=xi_(Gamma_(**))(1)\delta_{h, S}=\xi_{\Gamma_{*}}(1)δh,S=ξΓ∗(1) be the corresponding boundary divisor class.

3. PREVIOUS FORMULAS AND RESULTS

The first progress towards a formula for the double ramification cycle was when Faber and Pandharipande [7] proved that the double ramification cycle lies in the tautological ring, and thus in theory must be expressible in terms of the generators described in
the previous section. Their proof, although in principle constructive, involves a complicated recursion and does not seem to yield a practical formula.
The first progress towards an explicit formula came when Hain [10] computed the double ramification cycle when restricted to the compact type locus M g , n c t M g , n c t M_(g,n)^(ct)\mathcal{M}_{g, n}^{\mathrm{ct}}Mg,nct. On this locus the double ramification cycle is the pullback along an Abel-Jacobi map j A : M g , n c t X g , n j A : M g , n c t → X g , n j_(A):M_(g,n)^(ct)rarrX_(g,n)j_{A}: \mathcal{M}_{g, n}^{\mathrm{ct}} \rightarrow \mathcal{X}_{g, n}jA:Mg,nct→Xg,n of the class of the zero section Z g , n Z g , n Z_(g,n)\mathcal{Z}_{g, n}Zg,n of the universal abelian variety X g , n A g , n X g , n → A g , n X_(g,n)rarrA_(g,n)\mathcal{X}_{g, n} \rightarrow \mathcal{A}_{g, n}Xg,n→Ag,n. Hain showed that the class of this zero section is [ Z g , n ] = Θ g / g Z g , n = Θ g / g [Z_(g,n)]=Theta^(g)//g\left[\mathcal{Z}_{g, n}\right]=\Theta^{g} / g[Zg,n]=Θg/g ! and computed the pullback of the theta divisor Θ Î˜ Theta\ThetaΘ as an explicit divisor on M g , n c t M g , n c t M_(g,n)^(ct)\mathcal{M}_{g, n}^{\mathrm{ct}}Mg,nct,
(3.1) j A Θ = i = 1 n a i 2 2 ψ i h , S a S 2 4 δ h , S (3.1) j A ∗ Θ = ∑ i = 1 n   a i 2 2 ψ i − ∑ h , S   a S 2 4 δ h , S {:(3.1)j_(A)^(**)Theta=sum_(i=1)^(n)(a_(i)^(2))/(2)psi_(i)-sum_(h,S)(a_(S)^(2))/(4)delta_(h,S):}\begin{equation*} j_{A}^{*} \Theta=\sum_{i=1}^{n} \frac{a_{i}^{2}}{2} \psi_{i}-\sum_{h, S} \frac{a_{S}^{2}}{4} \delta_{h, S} \tag{3.1} \end{equation*}(3.1)jA∗Θ=∑i=1nai22ψi−∑h,SaS24δh,S
where a S = i S a i a S = ∑ i ∈ S   a i a_(S)=sum_(i in S)a_(i)a_{S}=\sum_{i \in S} a_{i}aS=∑i∈Sai and the second sum runs over all h , S h , S h,Sh, Sh,S defining boundary divisor classes.
Hain's formula for the compact type double ramification cycle is then
DR g c t ( A ) = 1 g ! ( j A Θ ) g = 1 g ! ( i = 1 n a i 2 2 ψ i h , S a S 2 4 δ h , S ) g DR g c t ⁡ ( A ) = 1 g ! j A ∗ Θ g = 1 g ! ∑ i = 1 n   a i 2 2 ψ i − ∑ h , S   a S 2 4 δ h , S g DR_(g)^(ct)(A)=(1)/(g!)(j_(A)^(**)Theta)^(g)=(1)/(g!)(sum_(i=1)^(n)(a_(i)^(2))/(2)psi_(i)-sum_(h,S)(a_(S)^(2))/(4)delta_(h,S))^(g)\operatorname{DR}_{g}^{\mathrm{ct}}(A)=\frac{1}{g!}\left(j_{A}^{*} \Theta\right)^{g}=\frac{1}{g!}\left(\sum_{i=1}^{n} \frac{a_{i}^{2}}{2} \psi_{i}-\sum_{h, S} \frac{a_{S}^{2}}{4} \delta_{h, S}\right)^{g}DRgct⁡(A)=1g!(jA∗Θ)g=1g!(∑i=1nai22ψi−∑h,SaS24δh,S)g
The divisor formula (3.1) is a homogeneous polynomial of degree 2 in A A AAA, so Hain's DR formula (3.2) is a homogeneous polynomial of degree 2 g 2 g 2g2 g2g in A A AAA.
Grushevsky and Zakharov [9] extended Hain's computation slightly, expanding from M g , n c t M g , n c t M_(g,n)^(ct)\mathcal{M}_{g, n}^{\mathrm{ct}}Mg,nct to a slightly larger open subscheme of M ¯ g , n M ¯ g , n bar(M)_(g,n)\overline{\mathcal{M}}_{g, n}M¯g,n by adding the locus of curves whose dual graph is a tree with a single loop added at one vertex. If Γ Î“ Gamma\GammaΓ is the stable graph with a single vertex and single loop, then their correction term is the codimension g g ggg part of
(3.3) ξ Γ ( i = 1 n exp ( 1 2 a i 2 ψ i ) k = 1 B 2 k 2 k k ! ( ψ + ψ ) k 1 ) (3.3) ξ Γ ∗ − ∏ i = 1 n   exp ⁡ 1 2 a i 2 ψ i ∑ k = 1 ∞   B 2 k 2 k k ! ψ + ψ ′ k − 1 {:(3.3)xi_(Gamma**)(-prod_(i=1)^(n)exp((1)/(2)a_(i)^(2)psi_(i))sum_(k=1)^(oo)(B_(2k))/(2^(k)k!)(psi+psi^('))^(k-1)):}\begin{equation*} \xi_{\Gamma *}\left(-\prod_{i=1}^{n} \exp \left(\frac{1}{2} a_{i}^{2} \psi_{i}\right) \sum_{k=1}^{\infty} \frac{B_{2 k}}{2^{k} k!}\left(\psi+\psi^{\prime}\right)^{k-1}\right) \tag{3.3} \end{equation*}(3.3)ξΓ∗(−∏i=1nexp⁡(12ai2ψi)∑k=1∞B2k2kk!(ψ+ψ′)k−1)
where ψ 1 , , ψ n ψ 1 , … , ψ n psi_(1),dots,psi_(n)\psi_{1}, \ldots, \psi_{n}ψ1,…,ψn are the psi classes on the legs, ψ , ψ ψ , ψ ′ psi,psi^(')\psi, \psi^{\prime}ψ,ψ′ are the psi classes on the two halfedges of the loop, and B 2 k B 2 k B_(2k)B_{2 k}B2k is a Bernoulli number.
In particular, the double ramification cycle is no longer a homogeneous polynomial in A A AAA when computed beyond compact type. This was also seen in work of Buryak, Shadrin, Spitz, and Zvonkine [3], who showed that the top degree intersections of double ramification cycles with monomials in the psi classes are inhomogeneous polynomials of degree 2 g 2 g 2g2 g2g in A A AAA.

4. MOTIVATION FOR THE FORMULA

In this section we discuss various observations and ideas that come about when
one tries to extend Hain's formula (3.2) to M ¯ g , n M ¯ g , n bar(M)_(g,n)\overline{\mathcal{M}}_{g, n}M¯g,n to obtain a full double ramification cycle formula.

4.1. Expanding Hain's formula

Exponentiating a boundary divisor class can be done using the multiplication laws for tautological classes [8, APPENDIX A]. Multiplying out Hain's formula (3.2) in this way gives
a nice sum over trees: DR g c t ( A ) DR g c t ⁡ ( A ) DR_(g)^(ct)(A)\operatorname{DR}_{g}^{c t}(A)DRgct⁡(A) is the codimension g g ggg part of
T stable tree 1 | Aut ( T ) | ( ξ T ) [ i = 1 n exp ( 1 2 a i 2 ψ h i ) (1.1) e = { h , h } E ( T ) 1 exp ( 1 2 w ( h ) w ( h ) ( ψ h + ψ h ) ) ψ h + ψ h ] ∑ T  stable tree    1 | Aut ⁡ ( T ) | ξ T ∗ ∏ i = 1 n   exp ⁡ 1 2 a i 2 ψ h i (1.1) â‹… ∏ e = h , h ′ ∈ E ( T )   1 − exp ⁡ − 1 2 w ( h ) w h ′ ψ h + ψ h ′ ψ h + ψ h ′ {:[sum_(T" stable tree ")(1)/(|Aut(T)|)(xi_(T))_(**)[prod_(i=1)^(n)exp((1)/(2)a_(i)^(2)psi_(h_(i))):}],[(1.1){:*prod_(e={h,h^(')}in E(T))(1-exp(-(1)/(2)w(h)w(h^('))(psi_(h)+psi_(h^(')))))/(psi_(h)+psi_(h^(')))]]:}\begin{align*} \sum_{T \text { stable tree }} \frac{1}{|\operatorname{Aut}(T)|}\left(\xi_{T}\right)_{*}\left[\prod_{i=1}^{n} \exp \left(\frac{1}{2} a_{i}^{2} \psi_{h_{i}}\right)\right. \\ \left.\cdot \prod_{e=\left\{h, h^{\prime}\right\} \in E(T)} \frac{1-\exp \left(-\frac{1}{2} w(h) w\left(h^{\prime}\right)\left(\psi_{h}+\psi_{h^{\prime}}\right)\right)}{\psi_{h}+\psi_{h^{\prime}}}\right] \tag{1.1} \end{align*}∑T stable tree 1|Aut⁡(T)|(ξT)∗[∏i=1nexp⁡(12ai2ψhi)(1.1)⋅∏e={h,h′}∈E(T)1−exp⁡(−12w(h)w(h′)(ψh+ψh′))ψh+ψh′]
where the function w : H ( T ) Z w : H ( T ) → Z w:H(T)rarrZw: H(T) \rightarrow \mathbb{Z}w:H(T)→Z is defined here by contracting all the edges in the tree T T TTT other than the one containing h h hhh and then letting w ( h ) w ( h ) w(h)w(h)w(h) be the sum of the a i a i a_(i)a_{i}ai for the legs i i iii on the same vertex as the half-edge h h hhh.
Extending this formula to M ¯ g , n M ¯ g , n bar(M)_(g,n)\overline{\mathcal{M}}_{g, n}M¯g,n requires us to provide a polynomial (or power series) in the psi classes for every stable graph Γ Î“ Gamma\GammaΓ, not just every stable tree. The w ( h ) w ( h ) w(h)w(h)w(h) definition above does not naturally extend to non-separating edges, so it is not immediately clear how to do this. Moreover, we know that this power series needs to be (3.3) for the single-loop graph, so something quite new is going on even there.

4.2. Cohomological field theory axioms

A cohomological field theory (CohFT) is a collection of classes Ω g , n ( γ 1 , , γ n ) Ω g , n γ 1 , … , γ n Omega_(g,n)(gamma_(1),dots,gamma_(n))\Omega_{g, n}\left(\gamma_{1}, \ldots, \gamma_{n}\right)Ωg,n(γ1,…,γn) on M ¯ g , n M ¯ g , n bar(M)_(g,n)\overline{\mathcal{M}}_{g, n}M¯g,n for all g g ggg and n n nnn, where the inputs γ i γ i gamma_(i)\gamma_{i}γi belong to some finite set S S SSS (a basis for the state space of the CohFT). These classes must satisfy certain compatibility axioms relating them to each other under pullback by natural maps between the M ¯ g , n M ¯ g , n bar(M)_(g,n)\overline{\mathcal{M}}_{g, n}M¯g,n. For one basic treatment of CohFTs and Givental's R-matrix action, see [14]. The double ramification cycle is not quite a CohFT, but it satisfies some subset of the properties of one. For example, if j : M ¯ g 1 , n 1 + 1 × M ¯ g 2 , n 2 + 1 M ¯ g , n j : M ¯ g 1 , n 1 + 1 × M ¯ g 2 , n 2 + 1 → M ¯ g , n j: bar(M)_(g_(1),n_(1)+1)xx bar(M)_(g_(2),n_(2)+1)rarr bar(M)_(g,n)j: \overline{\mathcal{M}}_{g_{1}, n_{1}+1} \times \overline{\mathcal{M}}_{g_{2}, n_{2}+1} \rightarrow \overline{\mathcal{M}}_{g, n}j:M¯g1,n1+1×M¯g2,n2+1→M¯g,n is a separating gluing map where the marked points split into sets I 1 , I 2 I 1 , I 2 I_(1),I_(2)I_{1}, I_{2}I1,I2 with | I i | = n i I i = n i |I_(i)|=n_(i)\left|I_{i}\right|=n_{i}|Ii|=ni, then we have
j DR g ( a 1 , , a n ) = DR g 1 ( { a i i I 1 } , t ) DR g 2 ( { a i i I 2 } , t ) j ∗ DR g ⁡ a 1 , … , a n = DR g 1 ⁡ a i ∣ i ∈ I 1 , t ⊗ DR g 2 ⁡ a i ∣ i ∈ I 2 , − t j^(**)DR_(g)(a_(1),dots,a_(n))=DR_(g_(1))({a_(i)∣i inI_(1)},t)oxDR_(g_(2))({a_(i)∣i inI_(2)},-t)j^{*} \operatorname{DR}_{g}\left(a_{1}, \ldots, a_{n}\right)=\operatorname{DR}_{g_{1}}\left(\left\{a_{i} \mid i \in I_{1}\right\}, t\right) \otimes \operatorname{DR}_{g_{2}}\left(\left\{a_{i} \mid i \in I_{2}\right\},-t\right)j∗DRg⁡(a1,…,an)=DRg1⁡({ai∣i∈I1},t)⊗DRg2⁡({ai∣i∈I2},−t)
where t Z t ∈ Z t inZt \in \mathbb{Z}t∈Z is the unique insertion that makes the parameters sum to 0 in each DR term on the right.
If the double ramification cycle were a CohFT, we would want a similar formula for the pullback along the nonseparating gluing map k : M ¯ g 1 , n + 2 M ¯ g , n k : M ¯ g − 1 , n + 2 → M ¯ g , n k: bar(M)_(g-1,n+2)rarr bar(M)_(g,n)k: \overline{\mathcal{M}}_{g-1, n+2} \rightarrow \overline{\mathcal{M}}_{g, n}k:M¯g−1,n+2→M¯g,n : the natural thing to write down would be
k DR g ( a 1 , , a n ) = t Z DR g 1 ( a 1 , , a n , t , t ) k ∗ DR g ⁡ a 1 , … , a n = ∑ t ∈ Z   DR g − 1 ⁡ a 1 , … , a n , t , − t k^(**)DR_(g)(a_(1),dots,a_(n))=sum_(t inZ)DR_(g-1)(a_(1),dots,a_(n),t,-t)k^{*} \operatorname{DR}_{g}\left(a_{1}, \ldots, a_{n}\right)=\sum_{t \in \mathbb{Z}} \operatorname{DR}_{g-1}\left(a_{1}, \ldots, a_{n}, t,-t\right)k∗DRg⁡(a1,…,an)=∑t∈ZDRg−1⁡(a1,…,an,t,−t)
but it is not clear how one might make sense of this infinite sum-it will not converge in any standard sense. What is going wrong here is that CohFTs are supposed to depend multilinearly on parameters from a finite-dimensional state space, but double ramification cycles take inputs in Z Z Z\mathbb{Z}Z so the state space appears to be infinite-dimensional.
So the double ramification cycle behaves like a CohFT as far as separating nodes are concerned, but the state space would have to be infinite-dimensional and this makes it unclear what to do at nonseparating nodes.

4.3. Givental's R-matrix action

Teleman [16] proved that semisimple CohFTs all have a very particular graph sum form, given by applying Givental's R-matrix action to a CohFT that lives fully in codimension zero. The rough shape of the resulting formula for a semisimple CohFT is
Ω g , n ( γ 1 , , γ n ) = Γ stable graph w : H ( Γ ) S 1 | Aut ( Γ ) | ( ξ Γ ) [ v V ( Γ ) ( vertex factor ) i = 1 n ( leg factor ) e = { h , h } E ( Γ ) ( edge factor ) ] Ω g , n γ 1 , … , γ n = ∑ Γ  stable graph    ∑ w : H ( Γ ) → S   1 | Aut ⁡ ( Γ ) | ξ Γ ∗ [ ∏ v ∈ V ( Γ )   (  vertex factor  ) ∏ i = 1 n   (  leg factor  ) ∏ e = h , h ′ ∈ E ( Γ )   (  edge factor  ) {:[Omega_(g,n)(gamma_(1),dots,gamma_(n))=sum_(Gamma" stable graph ")sum_(w:H(Gamma)rarr S)(1)/(|Aut(Gamma)|)(xi_(Gamma))_(**)[],[{:prod_(v in V(Gamma))(" vertex factor ")prod_(i=1)^(n)(" leg factor ")prod_(e={h,h^(')}in E(Gamma))(" edge factor ")]]:}\begin{aligned} \Omega_{g, n}\left(\gamma_{1}, \ldots, \gamma_{n}\right)= & \sum_{\Gamma \text { stable graph }} \sum_{w: H(\Gamma) \rightarrow S} \frac{1}{|\operatorname{Aut}(\Gamma)|}\left(\xi_{\Gamma}\right)_{*}[ \\ & \left.\prod_{v \in V(\Gamma)}(\text { vertex factor }) \prod_{i=1}^{n}(\text { leg factor }) \prod_{e=\left\{h, h^{\prime}\right\} \in E(\Gamma)}(\text { edge factor })\right] \end{aligned}Ωg,n(γ1,…,γn)=∑Γ stable graph ∑w:H(Γ)→S1|Aut⁡(Γ)|(ξΓ)∗[∏v∈V(Γ)( vertex factor )∏i=1n( leg factor )∏e={h,h′}∈E(Γ)( edge factor )]
where the second sum is over functions w w www on the half-edges of the graph taking values in some set S S SSS (a basis for the state space of the CohFT) and the values of w w www on the legs h 1 , , h n h 1 , … , h n h_(1),dots,h_(n)h_{1}, \ldots, h_{n}h1,…,hn are given as w ( h i ) = γ i w h i = γ i w(h_(i))=gamma_(i)w\left(h_{i}\right)=\gamma_{i}w(hi)=γi. The various factors are then power series (that depend on w w www ) in the corresponding kappa and psi classes. The expanded version of Hain's compact type formula (4.1) is of this shape: we take S = Z S = Z S=ZS=\mathbb{Z}S=Z, the vertex factor is 0 unless all of the incident w ( h ) w ( h ) w(h)w(h)w(h) sum to zero, and the edge factor is 0 unless the two w ( h ) w ( h ) w(h)w(h)w(h) along the edge sum to zero. These vanishings effectively place the following constraints on w w www (to get a nonzero contribution to D R g c t ( A ) ) D R g c t ( A ) {:DR_(g)^(ct)(A))\left.\mathrm{DR}_{g}^{\mathrm{ct}}(A)\right)DRgct(A)) :
(1) w ( h i ) = a i w h i = a i w(h_(i))=a_(i)w\left(h_{i}\right)=a_{i}w(hi)=ai for i = 1 , 2 , , n i = 1 , 2 , … , n i=1,2,dots,ni=1,2, \ldots, ni=1,2,…,n, where h i h i h_(i)h_{i}hi is the i i iii th leg;
(2) w ( h ) + w ( h ) = 0 w ( h ) + w h ′ = 0 w(h)+w(h^('))=0w(h)+w\left(h^{\prime}\right)=0w(h)+w(h′)=0 if { h , h } h , h ′ {h,h^(')}\left\{h, h^{\prime}\right\}{h,h′} is an edge;
(3) h v w ( h ) = 0 ∑ h → v   w ( h ) = 0 sum_(h rarr v)w(h)=0\sum_{h \rightarrow v} w(h)=0∑h→vw(h)=0 for each vertex v v vvv.
We say w w www is balanced (with respect to A A AAA ) if it satisfies these constraints. In other words, w w www is a flow on Γ Î“ Gamma\GammaΓ with sources/sinks at the legs (with specified values given there by A A AAA ). When Γ Î“ Gamma\GammaΓ is a tree, there is a unique such balanced w w www and we recover the w ( h ) w ( h ) w(h)w(h)w(h) used in (4.1).
From this perspective it is natural to just try to take (4.1) and extend it to be a Givental-type sum over arbitrary graphs (not just trees), but then there will be infinitely many choices of w w www and the resulting infinite sums will be nonconvergent. Moreover, careful comparison with the exact form of Givental's R-matrix action suggests that the vertex factor should contribute a total factor of something like " | Z | h 1 ( Γ ) | Z | − h 1 ( Γ ) |Z|^(-h^(1)(Gamma))|\mathbb{Z}|^{-h^{1}(\Gamma)}|Z|−h1(Γ)." Note that the set of balanced w w www is a torsor over H 1 ( Γ ; Z ) Z h 1 ( Γ ) H 1 ( Γ ; Z ) ≅ Z h 1 ( Γ ) H_(1)(Gamma;Z)~=Z^(h^(1)(Gamma))H_{1}(\Gamma ; \mathbb{Z}) \cong \mathbb{Z}^{h^{1}(\Gamma)}H1(Γ;Z)≅Zh1(Γ), so this factor feels like some sort of infinite averaging procedure.

4.4. Divergent averages

Returning to the simplest non-tree case, the graph with one vertex and one loop, matching things up with (3.3) would then require making sense of the "infinite average" identity
(4.2) 1 | Z | c Z c 2 k = B 2 k (4.2) 1 | Z | ∑ c ∈ Z   c 2 k = B 2 k {:(4.2)(1)/(|Z|)sum_(c inZ)c^(2k)=B_(2k):}\begin{equation*} \frac{1}{|\mathbb{Z}|} \sum_{c \in \mathbb{Z}} c^{2 k}=B_{2 k} \tag{4.2} \end{equation*}(4.2)1|Z|∑c∈Zc2k=B2k
This is reminiscent of the zeta regularization sum
c 1 c 2 k 1 = ζ ( 1 2 k ) = B 2 k 2 k ∑ c ≥ 1   c 2 k − 1 = ζ ( 1 − 2 k ) = − B 2 k 2 k sum_(c >= 1)c^(2k-1)=zeta(1-2k)=-(B_(2k))/(2k)\sum_{c \geq 1} c^{2 k-1}=\zeta(1-2 k)=-\frac{B_{2 k}}{2 k}∑c≥1c2k−1=ζ(1−2k)=−B2k2k
but there is no obvious way to make sense of this similarity. Moreover, more complicated graphs require much more complicated divergent sums; for example, a graph with two vertices, a double edge between them, and one leg on each vertex gives rise to infinite sums like
(4.3) 1 | Z | c + d = a c 2 k d 2 l (4.3) 1 | Z | ∑ c + d = a   c 2 k d 2 l {:(4.3)(1)/(|Z|)sum_(c+d=a)c^(2k)d^(2l):}\begin{equation*} \frac{1}{|\mathbb{Z}|} \sum_{c+d=a} c^{2 k} d^{2 l} \tag{4.3} \end{equation*}(4.3)1|Z|∑c+d=ac2kd2l
which must be interpreted.

4.5. Interpolating finite rank CohFTs

The problem with writing down a double ramification cycle formula of this type is clearly that the state space is infinite-dimensional. If we replace Z Z Z\mathbb{Z}Z with Z / r Z Z / r Z Z//rZ\mathbb{Z} / r \mathbb{Z}Z/rZ everywhere then there is no difficulty with writing down a similar-looking finite rank CohFT. The result might be something like the following (the case of a diagonal R-matrix-for an example of a more complicated CohFT of this general type, see [15]):
Γ stable graph w : H ( Γ ) Z / r Z balanced 1 | Aut ( Γ ) | ( ξ Γ ) [ 1 r h 1 ( Γ ) i = 1 n exp ( F w ( h i ) ( ψ h i ) ) e = { h , h } E ( Γ ) 1 exp ( F w ( h ) ( ψ h ) + F w ( h ) ( ψ h ) ) ψ h + ψ h ] , ∑ Γ  stable graph    ∑ w : H ( Γ ) → Z / r Z  balanced    1 | Aut ⁡ ( Γ ) | ξ Γ ∗ 1 r h 1 ( Γ ) ∏ i = 1 n   exp ⁡ F w h i ψ h i â‹… ∏ e = h , h ′ ∈ E ( Γ )   1 − exp ⁡ F w ( h ) ψ h + F w h ′ ψ h ′ ψ h + ψ h ′ , {:[sum_(Gamma" stable graph ")sum_({:[w:H(Gamma)rarrZ//rZ],[" balanced "]:})(1)/(|Aut(Gamma)|)(xi_(Gamma))_(**)[(1)/(r^(h^(1)(Gamma)))prod_(i=1)^(n)exp(F_(w(h_(i)))(psi_(h_(i)))):}],[{:*prod_(e={h,h^(')}in E(Gamma))(1-exp(F_(w(h))(psi_(h))+F_(w(h^(')))(psi_(h^(')))))/(psi_(h)+psi_(h^(')))]","]:}\begin{aligned} \sum_{\Gamma \text { stable graph }} \sum_{\substack{w: H(\Gamma) \rightarrow \mathbb{Z} / r \mathbb{Z} \\ \text { balanced }}} \frac{1}{|\operatorname{Aut}(\Gamma)|}\left(\xi_{\Gamma}\right)_{*}\left[\frac{1}{r^{h^{1}(\Gamma)}} \prod_{i=1}^{n} \exp \left(F_{w\left(h_{i}\right)}\left(\psi_{h_{i}}\right)\right)\right. \\ \left.\cdot \prod_{e=\left\{h, h^{\prime}\right\} \in E(\Gamma)} \frac{1-\exp \left(F_{w(h)}\left(\psi_{h}\right)+F_{w\left(h^{\prime}\right)}\left(\psi_{h^{\prime}}\right)\right)}{\psi_{h}+\psi_{h^{\prime}}}\right], \end{aligned}∑Γ stable graph ∑w:H(Γ)→Z/rZ balanced 1|Aut⁡(Γ)|(ξΓ)∗[1rh1(Γ)∏i=1nexp⁡(Fw(hi)(ψhi))⋅∏e={h,h′}∈E(Γ)1−exp⁡(Fw(h)(ψh)+Fw(h′)(ψh′))ψh+ψh′],
for power series F a ( Z ) F a ( Z ) F_(a)(Z)F_{a}(Z)Fa(Z) for a Z / r Z a ∈ Z / r Z a inZ//rZa \in \mathbb{Z} / r \mathbb{Z}a∈Z/rZ with F 0 ( Z ) = 0 F 0 ( Z ) = 0 F_(0)(Z)=0F_{0}(Z)=0F0(Z)=0 and F a ( Z ) = F a ( Z ) F − a ( − Z ) = − F a ( Z ) F_(-a)(-Z)=-F_(a)(Z)F_{-a}(-Z)=-F_{a}(Z)F−a(−Z)=−Fa(Z).
If we take F a ( Z ) = 1 2 a 2 Z F a ( Z ) = 1 2 a 2 Z F_(a)(Z)=(1)/(2)a^(2)ZF_{a}(Z)=\frac{1}{2} a^{2} ZFa(Z)=12a2Z for r 2 < a r 2 − r 2 < a ≤ r 2 -(r)/(2) < a <= (r)/(2)-\frac{r}{2}<a \leq \frac{r}{2}−r2<a≤r2 then this CohFT starts to look very much like the expanded version of Hain's formula, (4.1). In fact, if Γ Î“ Gamma\GammaΓ is a tree then the Γ Î“ Gamma\GammaΓ-term in this sum agrees with that in Hain's formula for all sufficiently large r r rrr. So it is tempting to try to take the limit as r r → ∞ r rarr oor \rightarrow \inftyr→∞ of these CohFTs. But this is not quite right: the r r rrr-version of the left side of (4.2) is then
1 r r 2 < c r 2 c 2 k 1 r ∑ − r 2 < c ≤ r 2   c 2 k (1)/(r)sum_(-(r)/(2) < c <= (r)/(2))c^(2k)\frac{1}{r} \sum_{-\frac{r}{2}<c \leq \frac{r}{2}} c^{2 k}1r∑−r2<c≤r2c2k
This certainly does not converge as r r → ∞ r rarr oor \rightarrow \inftyr→∞. However, if we restrict to even r r rrr then it is polynomial in r r rrr, and if we examine the coefficients of this polynomial then we see that B 2 k B 2 k B_(2k)B_{2 k}B2k, the desired value, is the constant coefficient in r r rrr.
This suggests a potential interpretation even of more complicated sums like (4.3):
1 | Z | c + d = a c 2 k d 2 l = 1 | Z / 0 Z | c , d Z / 0 Z c + d = a ( mod 0 ) c 2 k d 2 l := [ 1 | Z / r Z | c , d Z / r Z c + d = a ( mod r ) c 2 k d 2 l ] r = 0 1 | Z | ∑ c + d = a   c 2 k d 2 l = 1 | Z / 0 Z | ∑ c , d ∈ Z / 0 Z c + d = a ( mod 0 )   c 2 k d 2 l := 1 | Z / r Z | ∑ c , d ∈ Z / r Z c + d = a ( mod r )   c 2 k d 2 l r = 0 {:[(1)/(|Z|)sum_(c+d=a)c^(2k)d^(2l)=(1)/(|Z//0Z|)sum_({:[c","d inZ//0Z],[c+d=a(mod0)]:})c^(2k)d^(2l)],[:=[(1)/(|Z//rZ|)sum_({:[c","d inZ//rZ],[c+d=a(mod r)]:})c^(2k)d^(2l)]_(r=0)]:}\begin{aligned} \frac{1}{|\mathbb{Z}|} \sum_{c+d=a} c^{2 k} d^{2 l} & =\frac{1}{|\mathbb{Z} / 0 \mathbb{Z}|} \sum_{\begin{array}{c} c, d \in \mathbb{Z} / 0 \mathbb{Z} \\ c+d=a(\bmod 0) \end{array}} c^{2 k} d^{2 l} \\ & :=\left[\frac{1}{|\mathbb{Z} / r \mathbb{Z}|} \sum_{\substack{c, d \in \mathbb{Z} / r \mathbb{Z} \\ c+d=a(\bmod r)}} c^{2 k} d^{2 l}\right]_{r=0} \end{aligned}1|Z|∑c+d=ac2kd2l=1|Z/0Z|∑c,d∈Z/0Zc+d=a(mod0)c2kd2l:=[1|Z/rZ|∑c,d∈Z/rZc+d=a(modr)c2kd2l]r=0
where c c ccc and d d ddd must be interpreted inside c 2 k d 2 l c 2 k d 2 l c^(2k)d^(2l)c^{2 k} d^{2 l}c2kd2l as elements of Z Z Z\mathbb{Z}Z via some choice of mod r mod r mod r\bmod rmodr representatives (we used r / 2 + 1 , , r / 2 − r / 2 + 1 , … , r / 2 -r//2+1,dots,r//2-r / 2+1, \ldots, r / 2−r/2+1,…,r/2 before but 0 , , r 1 0 , … , r − 1 0,dots,r-10, \ldots, r-10,…,r−1 will give the same final answer) and setting r = 0 r = 0 r=0r=0r=0 at the end is done by polynomial interpolation.

4.6. Geometric interpretation from ( k / r ) ( k / r ) (k//r)(k / r)(k/r)-spin structures

An ( k / r ) ( k / r ) (k//r)(k / r)(k/r)-spin structure on a smooth curve C C CCC with marked points p i p i p_(i)p_{i}pi and weights a i a i a_(i)a_{i}ai is a choice of line bundle L L LLL on C C CCC such that L r ω C k ( a 1 p 1 + + a n p n ) L ⊗ r ≡ ω C ⊗ k a 1 p 1 + ⋯ + a n p n L^(ox r)-=omega_(C)^(ox k)(a_(1)p_(1)+cdots+a_(n)p_(n))L^{\otimes r} \equiv \omega_{C}^{\otimes k}\left(a_{1} p_{1}+\cdots+a_{n} p_{n}\right)L⊗r≡ωC⊗k(a1p1+⋯+anpn). If we take k = 0 k = 0 k=0k=0k=0 and assume the weights a i a i a_(i)a_{i}ai sum to zero then for any positive r r rrr any smooth curve will have such " r r rrr th root structures." But we can also interpret this construction as meaningful when r = 0 r = 0 r=0r=0r=0, when we get that a curve only admits a (0/0)-spin structure if it is in the double ramification locus. This observation gives a vague geometric idea for what it might mean to think of the double ramification cycle as given by specializing some parameter r r rrr to 0 .

5. THE DOUBLE RAMIFICATION CYCLE FORMULA

We can now state the main result of [11], the double ramification cycle formula:
Theorem 1 ([11]). D R g ( A ) D R g ( A ) DR_(g)(A)\mathrm{DR}_{g}(A)DRg(A) is the codimension g g ggg part of
Γ stable graph 1 | Aut ( Γ ) | ( ξ Γ ) [ 1 | Z | h 1 ( Γ ) w : H ( Γ ) Z balanced i = 1 n exp ( 1 2 a i 2 ψ h i ) e = { h , h } E ( Γ ) 1 exp ( 1 2 w ( h ) w ( h ) ( ψ h + ψ h ) ) ψ h + ψ h ] ∑ Γ  stable graph    1 | Aut ⁡ ( Γ ) | ξ Γ ∗ 1 | Z | h 1 ( Γ ) ∑ w : H ( Γ ) → Z  balanced    ∏ i = 1 n   exp ⁡ 1 2 a i 2 ψ h i â‹… ∏ e = h , h ′ ∈ E ( Γ )   1 − exp ⁡ − 1 2 w ( h ) w h ′ ψ h + ψ h ′ ψ h + ψ h ′ {:[sum_(Gamma" stable graph ")(1)/(|Aut(Gamma)|)(xi_(Gamma))_(**)[(1)/(|Z|^(h^(1)(Gamma)))sum_({:[w:H(Gamma)rarrZ],[" balanced "]:})prod_(i=1)^(n)exp((1)/(2)a_(i)^(2)psi_(h_(i))):}],[{:*prod_(e={h,h^(')}in E(Gamma))(1-exp(-(1)/(2)w(h)w(h^('))(psi_(h)+psi_(h^(')))))/(psi_(h)+psi_(h^(')))]]:}\begin{aligned} \sum_{\Gamma \text { stable graph }} \frac{1}{|\operatorname{Aut}(\Gamma)|}\left(\xi_{\Gamma}\right)_{*}\left[\frac{1}{|\mathbb{Z}|^{h^{1}(\Gamma)}} \sum_{\begin{array}{c} w: H(\Gamma) \rightarrow \mathbb{Z} \\ \text { balanced } \end{array}} \prod_{i=1}^{n} \exp \left(\frac{1}{2} a_{i}^{2} \psi_{h_{i}}\right)\right. \\ \left.\cdot \prod_{e=\left\{h, h^{\prime}\right\} \in E(\Gamma)} \frac{1-\exp \left(-\frac{1}{2} w(h) w\left(h^{\prime}\right)\left(\psi_{h}+\psi_{h^{\prime}}\right)\right)}{\psi_{h}+\psi_{h^{\prime}}}\right] \end{aligned}∑Γ stable graph 1|Aut⁡(Γ)|(ξΓ)∗[1|Z|h1(Γ)∑w:H(Γ)→Z balanced ∏i=1nexp⁡(12ai2ψhi)⋅∏e={h,h′}∈E(Γ)1−exp⁡(−12w(h)w(h′)(ψh+ψh′))ψh+ψh′]
where formal expressions of the form
1 | Z | h 1 ( Γ ) w : H ( Γ ) Z balanced P ( { w ( h ) } ) 1 | Z | h 1 ( Γ ) ∑ w : H ( Γ ) → Z  balanced    P ( { w ( h ) } ) (1)/(|Z|^(h^(1)(Gamma)))sum_({:[w:H(Gamma)rarrZ],[" balanced "]:})P({w(h)})\frac{1}{|\mathbb{Z}|^{h^{1}(\Gamma)}} \sum_{\substack{w: H(\Gamma) \rightarrow \mathbb{Z} \\ \text { balanced }}} P(\{w(h)\})1|Z|h1(Γ)∑w:H(Γ)→Z balanced P({w(h)})
(for P P PPP a polynomial) are evaluated by setting r = 0 r = 0 r=0r=0r=0 in the corresponding r r rrr-polynomial
1 r h 1 ( Γ ) w : H ( Γ ) { 0 , 1 , , r 1 } balanced mod r P ( { w ( h ) } ) 1 r h 1 ( Γ ) ∑ w : H ( Γ ) → { 0 , 1 , … , r − 1 }  balanced mod  r   P ( { w ( h ) } ) (1)/(r^(h^(1)(Gamma)))sum_({:[w:H(Gamma)rarr{0","1","dots","r-1}],[" balanced mod "r]:})P({w(h)})\frac{1}{r^{h^{1}(\Gamma)}} \sum_{\substack{w: H(\Gamma) \rightarrow\{0,1, \ldots, r-1\} \\ \text { balanced mod } r}} P(\{w(h)\})1rh1(Γ)∑w:H(Γ)→{0,1,…,r−1} balanced mod rP({w(h)})
The combinatorial result (necessary for this theorem statement to make sense) that the expression in the final line is in fact a polynomial in r r rrr (for r r rrr sufficiently large) was proved in [11, APPENDIX A].
The proof of Theorem 1 in [11] follows some of the motivation in Section 4. We first explain the meaning of the additional r r rrr parameter. For each r > 0 r > 0 r > 0r>0r>0, let P 1 [ r ] P 1 [ r ] P^(1)[r]\mathbb{P}^{1}[r]P1[r] denote the projective line with a B Z r B Z r BZ_(r)B \mathbb{Z}_{r}BZr orbifold point at 0 . One can then use C C ∗ C^(**)\mathbb{C}^{*}C∗-localization on the moduli space of relative stable maps to P 1 [ r ] / { } P 1 [ r ] / { ∞ } P^(1)[r]//{oo}\mathbb{P}^{1}[r] /\{\infty\}P1[r]/{∞} to obtain complicated relations that entangle double ramification cycles, classes coming from moduli of ( 0 / r ) ( 0 / r ) (0//r)(0 / r)(0/r)-spin curves (discussed briefly in the case of smooth curves in Section 4.6), and other basic tautological classes. The relevant ( 0 / r ) ( 0 / r ) (0//r)(0 / r)(0/r)-spin classes were previously computed by Chiodo [4] using GrothendieckRiemann-Roch.
These localization relations are too difficult to study effectively for specific values of r r rrr, but it turns out that they have polynomial dependence on r r rrr. Taking the constant term in r r rrr simplifies them greatly: most of the terms vanish, and the only remaining terms are a
single double ramification cycle and the r = 0 r = 0 r=0r=0r=0 interpolation of certain classes written in terms of the Chern characters of the pushforward of the universal r r rrr th root line bundle on the moduli space of ( 0 / r ) ( 0 / r ) (0//r)(0 / r)(0/r)-spin curves. Chiodo's formula [4] for these Chern characters gives that these classes are CohFTs with formulas of the type described in Section 4.5. The power series in psi classes appearing in these formulas do not look exactly like those appearing in Theorem 1, but they have the same r = 0 r = 0 r=0r=0r=0 interpolation. (In the language of Section 4.5, the power series F a ( Z ) F a ( Z ) F_(a)(Z)F_{a}(Z)Fa(Z) will be congruent to 1 2 a 2 Z mod r 1 2 a 2 Z mod r (1)/(2)a^(2)Z mod r\frac{1}{2} a^{2} Z \bmod r12a2Zmodr.) The result is a proof of Theorem 1 .

ACKNOWLEDGMENTS

The author is grateful to R. Cavalieri, D. Holmes, F. Janda, S. Grushevsky, S. Molcho, G. Oberdieck, R. Pandharipande, J. Schmitt, D. Zakharov, and D. Zvonkine for many discussions about double ramification cycles and related topics over the years.

FUNDING

The author was partially supported by National Science Foundation Grant No. 1807079.

REFERENCES

[1] E. Arbarello and M. Cornalba, Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves. J. Algebraic Geom. 5 (1996), no. 4, 705-749.
[2] Y. Bae, D. Holmes, R. Pandharipande, J. Schmitt, and R. Schwarz, Pixton's formula and Abel-Jacobi theory on the Picard stack. 2021, arXiv:2004.08676v2.
[3] A. Buryak, S. Shadrin, L. Spitz, and D. Zvonkine, Integrals of ψ ψ psi\psiψ-classes over double ramification cycles. Amer. J. Math. 137 (2015), no. 3, 699-737.
[4] A. Chiodo, Towards an enumerative geometry of the moduli space of twisted curves and r r rrr th roots. Compos. Math. 144 (2008), no. 6, 1461-1496.
[5] E. Clader and F. Janda, Pixton's double ramification cycle relations. Geom. Topol. 22 (2018), no. 2, 1069-1108.
[6] C. Faber and R. Pandharipande, Logarithmic series and Hodge integrals in the tautological ring. Michigan Math. J. 48 (2000), 215-252.
[7] C. Faber and R. Pandharipande, Relative maps and tautological classes. J. Eur. Math. Soc. (JEMS) 7 (2005), no. 1, 13-49.
[8] T. Graber and R. Pandharipande, Constructions of nontautological classes on moduli spaces of curves. Michigan Math. J. 51 (2003), no. 1, 93-109.
[9] S. Grushevsky and D. Zakharov, The zero section of the universal semiabelian variety and the double ramification cycle. Duke Math. J. 163 (2014), no. 5, 953-982.
[10] R. Hain, Normal functions and the geometry of moduli spaces of curves. In Handbook of moduli. Vol. I, pp. 527-578, Adv. Lect. Math. (ALM) 24, Int. Press, Somerville, MA, 2013.
[11] F. Janda, R. Pandharipande, A. Pixton, and D. Zvonkine, Double ramification cycles on the moduli spaces of curves. Publ. Math. Inst. Hautes Études Sci. 125 (2017), no. 1, 221-266.
[12] F. Janda, R. Pandharipande, A. Pixton, and D. Zvonkine, Double ramification cycles with target varieties. J. Topol. 13 (2020), no. 4, 1725-1766.
[13] S. Marcus and J. Wise, Logarithmic compactification of the Abel-Jacobi section. Proc. Lond. Math. Soc. (3) 121 (2020), no. 5, 1207-1250.
[14] R. Pandharipande, A. Pixton, and D. Zvonkine, Relations on M ¯ g , n M ¯ g , n bar(M)_(g,n)\bar{M}_{g, n}M¯g,n via 3-spin structures. J. Amer. Math. Soc. 28 (2015), no. 1, 279-309.
[15] R. Pandharipande, A. Pixton, and D. Zvonkine, Tautological relations via r r rrr-spin structures. J. Algebraic Geom. 28 (2019), no. 3, 439-496.
[16] C. Teleman, The structure of 2D semi-simple field theories. Invent. Math. 188 (2012), no. 3, 525-588.

AARON PIXTON

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA, pixton@umich.edu

EFFECTIVE RESULTS IN THE THREE-DIMENSIONAL MINIMAL MODEL PROGRAM
YURI PROKHOROV

ABSTRACT

We give a brief review on recent developments in the three-dimensional minimal model program.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 14E30; Secondary 14J30, 14B05, 14E05

KEYWORDS

Minimal model, Mori contraction, terminal (canonical) singularity, flip, extremal ray
In this note we give a brief review on recent developments in the three-dimensional minimal model program (MMP for short). Certainly, this is not a complete survey of all advances in this area. For example, we do not discuss the minimal models of varieties of nonnegative Kodaira dimension, as well as applications to birational geometry and moduli spaces.
The aim of the MMP is to find a good representative in a fixed birational equivalence class of algebraic varieties. Starting with an arbitrary smooth projective variety, one can perform a finite number of certain elementary transformations, called divisorial contractions and flips, and at the end obtain a variety which is simpler in some sense. Most parts of the MMP are completed in arbitrary dimension. One of the basic remaining problems is the following:

Describe all the intermediate steps and the outcome of the MMP.

The MMP makes sense only in dimensions 2 ≥ 2 >= 2\geq 2≥2, and for surfaces it is classical and well known. So the first nontrivial case is the three-dimensional one. It turns out that to proceed with the MMP in dimension 3 ≥ 3 >= 3\geq 3≥3, one has to work with varieties admitting certain types of very mild, the so-called terminal, singularities. On the other hand, dimension 3 is the last dimension where one can expect effective results: in higher dimensions, classification results become very complicated and unreasonably long.
We will work over the field C C C\mathbb{C}C of complex numbers throughout. A variety is either an algebraic variety or a reduced complex space.

1. SINGULARITIES

Recall that a Weil divisor D D DDD on a normal variety is said to be Q Q Q\mathbb{Q}Q-Cartier if its multiple n D n D nDn DnD, for some n n nnn, is a Cartier divisor. For any morphism f : Y X f : Y → X f:Y rarr Xf: Y \rightarrow Xf:Y→X, the pull-back f D f ∗ D f^(**)Df^{*} Df∗D of a Q Q Q\mathbb{Q}Q-Cartier divisor D D DDD is well defined as a divisor with rational coefficients ( Q Q Q\mathbb{Q}Q-divisor). A variety X X XXX has Q Q Q\mathbb{Q}Q-factorial singularities if any Weil divisor on X X XXX is Q Q Q\mathbb{Q}Q-Cartier.
Definition 1.1. A normal algebraic variety (or an analytic space) X X XXX is said to have terminal (resp. canonical, log terminal, log canonical) singularities if the canonical Weil divisor K X K X K_(X)K_{X}KX is Q Q Q\mathbb{Q}Q-Cartier and, for any birational morphism f : Y X f : Y → X f:Y rarr Xf: Y \rightarrow Xf:Y→X, one can write
(1.1.1) K Y = f K X + a i E i (1.1.1) K Y = f ∗ K X + ∑ a i E i {:(1.1.1)K_(Y)=f^(**)K_(X)+suma_(i)E_(i):}\begin{equation*} K_{Y}=f^{*} K_{X}+\sum a_{i} E_{i} \tag{1.1.1} \end{equation*}(1.1.1)KY=f∗KX+∑aiEi
where E i E i E_(i)E_{i}Ei are all the exceptional divisors and a i > 0 a i > 0 a_(i) > 0a_{i}>0ai>0 (resp. a i 0 , a i > 1 , a i 1 a i ≥ 0 , a i > − 1 , a i ≥ − 1 a_(i) >= 0,a_(i) > -1,a_(i) >= -1a_{i} \geq 0, a_{i}>-1, a_{i} \geq-1ai≥0,ai>−1,ai≥−1 ) for all i i iii. The smallest positive m m mmm such that m K X m K X mK_(X)m K_{X}mKX is Cartier is called the Gorenstein index of X X XXX. Canonical singularities of index 1 are rational Gorenstein.
The class of terminal Q Q Q\mathbb{Q}Q-factorial singularities is the smallest class that is closed under the MMP. Canonical singularities are important because they appear in the canonical models of varieties of general type. A crucial observation is that terminal singularities lie in codimension 3 ≥ 3 >= 3\geq 3≥3. In particular, terminal surface singularities are smooth and terminal threefold singularities are isolated. Canonical singularities of surfaces are called D u V a l D u V a l DuVal\mathrm{Du} \mathrm{Val}DuVal
or rational double points. Any two-dimensional log terminal singularity is a quotient of a smooth surface germ by a finite group [32]. Terminal threefolds singularities were classified by M. Reid [65] and S. Mori [43].
Example. Let X C 4 X ⊂ C 4 X subC^(4)X \subset \mathbb{C}^{4}X⊂C4 be a hypersurface given by the equation
ϕ ( x 1 , x 2 , x 3 ) + x 4 ψ ( x 1 , , x 4 ) = 0 Ï• x 1 , x 2 , x 3 + x 4 ψ x 1 , … , x 4 = 0 phi(x_(1),x_(2),x_(3))+x_(4)psi(x_(1),dots,x_(4))=0\phi\left(x_{1}, x_{2}, x_{3}\right)+x_{4} \psi\left(x_{1}, \ldots, x_{4}\right)=0Ï•(x1,x2,x3)+x4ψ(x1,…,x4)=0
where ϕ = 0 Ï• = 0 phi=0\phi=0Ï•=0 is an equation of a D u Val D u Val DuVal\mathrm{Du} \operatorname{Val}DuVal (ADE) singularity. Then the singularity of X X XXX at 0 is canonical Gorenstein. It is terminal if and only if it is isolated. Singularities of this type are called cDV.
According to [65], any three-dimensional terminal singularity of index m > 1 m > 1 m > 1m>1m>1 is a quotient of an isolated cDV-singularity by the cyclic group μ m μ m mu_(m)\boldsymbol{\mu}_{m}μm of order m m mmm. More precisely, we have the following
Theorem 1.2 ([65]). Let ( X P ) ( X ∋ P ) (X∋P)(X \ni P)(X∋P) be an analytic germ of a three-dimensional terminal singularity of index m 1 m ≥ 1 m >= 1m \geq 1m≥1. Then there exist an isolated c D V c D V cDV\mathrm{cDV}cDV-singularity ( X P ) X ♯ ∋ P ♯ (X^(♯)∋P^(♯))\left(X^{\sharp} \ni P^{\sharp}\right)(X♯∋P♯) and a cyclic μ m μ m mu_(m)\mu_{m}μm-cover
π : ( X P ) ( X P ) Ï€ : X ♯ ∋ P ♯ ⟶ ( X ∋ P ) pi:(X^(♯)∋P^(♯))longrightarrow(X∋P)\pi:\left(X^{\sharp} \ni P^{\sharp}\right) \longrightarrow(X \ni P)Ï€:(X♯∋P♯)⟶(X∋P)
which is étale outside P P PPP.
The morphism π Ï€ pi\piÏ€ in the above theorem is called the index-one cover. A detailed classification of all possibilities for the equations of X C 4 X ♯ ⊂ C 4 X^(♯)subC^(4)X^{\sharp} \subset \mathbb{C}^{4}X♯⊂C4 and the actions of μ m μ m mu_(m)\boldsymbol{\mu}_{m}μm was obtained in [43] (see also [66]).
Example. Let the cyclic group μ m μ m mu_(m)\mu_{m}μm act on C n C n C^(n)\mathbb{C}^{n}Cn diagonally via
( x 1 , , x n ) ( ζ a 1 x 1 , , ζ a n x n ) , ζ = ζ m = exp ( 2 π i / m ) x 1 , … , x n ↦ ζ a 1 x 1 , … , ζ a n x n , ζ = ζ m = exp ⁡ ( 2 Ï€ i / m ) (x_(1),dots,x_(n))|->(zeta^(a_(1))x_(1),dots,zeta^(a_(n))x_(n)),quad zeta=zeta_(m)=exp(2pii//m)\left(x_{1}, \ldots, x_{n}\right) \mapsto\left(\zeta^{a_{1}} x_{1}, \ldots, \zeta^{a_{n}} x_{n}\right), \quad \zeta=\zeta_{m}=\exp (2 \pi \mathrm{i} / m)(x1,…,xn)↦(ζa1x1,…,ζanxn),ζ=ζm=exp⁡(2Ï€i/m)
Then we say that ( a 1 , , a n ) a 1 , … , a n (a_(1),dots,a_(n))\left(a_{1}, \ldots, a_{n}\right)(a1,…,an) is the collection of weights of the action. Assume that the action is free in codimension 1 . Then the quotient singularity C n / μ m 0 C n / μ m ∋ 0 C^(n)//mu_(m)∋0\mathbb{C}^{n} / \mu_{m} \ni 0Cn/μm∋0 is said to be of type 1 m ( a 1 , , a n ) 1 m a 1 , … , a n (1)/(m)(a_(1),dots,a_(n))\frac{1}{m}\left(a_{1}, \ldots, a_{n}\right)1m(a1,…,an). According to the criterion (see [66, THEOREM 4.11]), this singularity is terminal if and only if
i = 1 n k a i ¯ > m for k = 1 , , m 1 ∑ i = 1 n   k a i ¯ > m  for  k = 1 , … , m − 1 sum_(i=1)^(n) bar(ka_(i)) > m quad" for "k=1,dots,m-1\sum_{i=1}^{n} \overline{k a_{i}}>m \quad \text { for } k=1, \ldots, m-1∑i=1nkai¯>m for k=1,…,m−1
where - is the smallest residue mod m m mmm. In the threefold case this criterion has a very simple form: a quotient singularity C m / μ m C m / μ m C^(m)//mu_(m)\mathbb{C}^{m} / \mu_{m}Cm/μm is terminal if and only if it is of type 1 m ( 1 , 1 , a ) 1 m ( 1 , − 1 , a ) (1)/(m)(1,-1,a)\frac{1}{m}(1,-1, a)1m(1,−1,a), where gcd ( m , a ) = 1 gcd ⁡ ( m , a ) = 1 gcd(m,a)=1\operatorname{gcd}(m, a)=1gcd⁡(m,a)=1. This is a cyclic quotient terminal singularity.
Example ( [ 43 , 66 ] [ 43 , 66 ] [43,66][43,66][43,66] ). Let the cyclic group μ m μ m mu_(m)\boldsymbol{\mu}_{m}μm act on C 4 C 4 C^(4)\mathbb{C}^{4}C4 diagonally with weights ( 1 , 1 , a , 0 ) ( 1 , − 1 , a , 0 ) (1,-1,a,0)(1,-1, a, 0)(1,−1,a,0), where gcd ( m , a ) = 1 gcd ⁡ ( m , a ) = 1 gcd(m,a)=1\operatorname{gcd}(m, a)=1gcd⁡(m,a)=1. Then for a polynomial ϕ ( u , v ) Ï• ( u , v ) phi(u,v)\phi(u, v)Ï•(u,v), the singularity at 0 of the quotient
{ x 1 x 2 + ϕ ( x 3 m , x 4 ) = 0 } / μ m x 1 x 2 + Ï• x 3 m , x 4 = 0 / μ m {x_(1)x_(2)+phi(x_(3)^(m),x_(4))=0}//mu_(m)\left\{x_{1} x_{2}+\phi\left(x_{3}^{m}, x_{4}\right)=0\right\} / \mu_{m}{x1x2+Ï•(x3m,x4)=0}/μm
is terminal whenever it is isolated. The index of this singularity equals m m mmm.
As a consequence of the classification, we obtain that the local fundamental group of the (analytic) germ of a three-dimensional terminal singularity of index m m mmm is cyclic of order m m mmm :
(1.2.1) π 1 ( X { P } ) Z / m Z (1.2.1) Ï€ 1 ( X ∖ { P } ) ≃ Z / m Z {:(1.2.1)pi_(1)(X\\{P})≃Z//mZ:}\begin{equation*} \pi_{1}(X \backslash\{P\}) \simeq \mathbb{Z} / m \mathbb{Z} \tag{1.2.1} \end{equation*}(1.2.1)Ï€1(X∖{P})≃Z/mZ
In particular, for any Weil Q Q Q\mathbb{Q}Q-Cartier divisor D D DDD on X X XXX, its m m mmm th multiple m D m D mDm DmD is Cartier [32, LEMMA 5.1].
The class of canonical threefold singularities is much larger than the class of terminal ones. However, there are certain boundedness results. For example, it is known that the index of a strictly canonical isolated singularity is at most 6 [31].
The modern higher-dimensional MMP often works with pairs, and one needs to extend Definition 1.1 to a wider class of objects.
Definition. Let X X XXX be a normal variety and let B B BBB be an effective Q Q Q\mathbb{Q}Q-divisor on X X XXX. The pair ( X , B ) ( X , B ) (X,B)(X, B)(X,B) is said to be p l t p l t pltp l tplt (resp. lc) if K X + B K X + B K_(X)+BK_{X}+BKX+B is Q Q Q\mathbb{Q}Q-Cartier and, for any birational morphism f : Y X f : Y → X f:Y rarr Xf: Y \rightarrow Xf:Y→X, one can write
K Y + B Y = f ( K X + B ) + a i E i K Y + B Y = f ∗ K X + B + ∑ a i E i K_(Y)+B_(Y)=f^(**)(K_(X)+B)+suma_(i)E_(i)K_{Y}+B_{Y}=f^{*}\left(K_{X}+B\right)+\sum a_{i} E_{i}KY+BY=f∗(KX+B)+∑aiEi
where B Y B Y B_(Y)B_{Y}BY is the proper transform of B , E i B , E i B,E_(i)B, E_{i}B,Ei are all the exceptional divisors and a i > 1 a i > − 1 a_(i) > -1a_{i}>-1ai>−1 (resp. a i 1 ) a i ≥ − 1 {:a_(i) >= -1)\left.a_{i} \geq-1\right)ai≥−1) for all i i iii. The pair ( X , B ) ( X , B ) (X,B)(X, B)(X,B) is said to be k l t k l t kltk l tklt if it is plt and B = 0 ⌊ B ⌋ = 0 |__ B __|=0\lfloor B\rfloor=0⌊B⌋=0.

2. MINIMAL MODEL PROGRAM

Basic elementary operations in the MMP are Mori contractions.
A contraction is a proper surjective morphism f : X Z f : X → Z f:X rarr Zf: X \rightarrow Zf:X→Z of normal varieties with connected fibers. The exceptional locus of a contraction f f fff is the subset Exc ( f ) X Exc ⁡ ( f ) ⊂ X Exc(f)sub X\operatorname{Exc}(f) \subset XExc⁡(f)⊂X of points at which f f fff is not an isomorphism. A Mori contraction is a contraction f : X Z f : X → Z f:X rarr Zf: X \rightarrow Zf:X→Z such that the variety X X XXX has at worst terminal Q Q Q\mathbb{Q}Q-factorial singularities, the anticanonical class K X − K X -K_(X)-K_{X}−KX is f f fff-ample, and the relative Picard number ρ ( X / Z ) ρ ( X / Z ) rho(X//Z)\rho(X / Z)ρ(X/Z) equals 1. A Mori contraction is said to be divisorial (resp. flipping) if it is birational and the locus Exc ( f ) Exc ⁡ ( f ) Exc(f)\operatorname{Exc}(f)Exc⁡(f) has codimension 1 (resp. 2 ≥ 2 >= 2\geq 2≥2 ). For a divisorial contraction, the exceptional locus Exc ( f ) Exc ⁡ ( f ) Exc(f)\operatorname{Exc}(f)Exc⁡(f) is a prime divisor. A Mori contraction whose target is a lower-dimensional variety is called a Mori fiber space. Then the general fiber is a Fano variety with at worst terminal singularities. In the particular cases where the relative dimension of X / Z X / Z X//ZX / ZX/Z equals 1 (resp. 2), the Mori fiber space f : X Z f : X → Z f:X rarr Zf: X \rightarrow Zf:X→Z is called a Q Q Q\mathbb{Q}Q-conic bundle (resp. Q Q Q\mathbb{Q}Q-del Pezzo fibration). If Z Z ZZZ is a point, then X X XXX is a Fano variety with at worst terminal Q Q Q\mathbb{Q}Q-factorial singularities and Pic ( X ) Z Pic ⁡ ( X ) ≃ Z Pic(X)≃Z\operatorname{Pic}(X) \simeq \mathbb{Z}Pic⁡(X)≃Z. For short, we call such varieties Q Q Q\mathbb{Q}Q-Fano.
The MMP procedure is a sequence of elementary transformations which are constructed inductively [ 35 , 39 ] [ 35 , 39 ] [35,39][35,39][35,39]. Let X X XXX be a projective algebraic variety with terminal Q Q Q\mathbb{Q}Q-factorial singularities. If the canonical divisor K X K X K_(X)K_{X}KX is not nef, then there exists a Mori contraction f : X Z f : X → Z f:X rarr Zf: X \rightarrow Zf:X→Z. If f f fff is divisorial, then Z Z ZZZ is again a variety with terminal Q Q Q\mathbb{Q}Q-factorial singularities and, in this situation, we can proceed with the MMP replacing X X XXX with Z Z ZZZ. In contrast,
a flipping contraction takes us out the category of terminal Q Q Q\mathbb{Q}Q-factorial varieties. To proceed, one has to perform a surgery operation as follows:
where f + f + f^(+)f^{+}f+is a contraction whose exceptional locus has codimension 2 ≥ 2 >= 2\geq 2≥2 and the divisor K X + K X + K_(X^(+))K_{X^{+}}KX+is Q Q Q\mathbb{Q}Q-Cartier and f + f + f^(+)f^{+}f+-ample. Then the variety X + X + X^(+)X^{+}X+again has terminal Q Q Q\mathbb{Q}Q-factorial singularities, and we can proceed by replacing X X XXX with X + X + X^(+)X^{+}X+.
The process described above should terminate, and at the end we obtain a variety X ¯ X ¯ bar(X)\bar{X}X¯ such that either X ¯ X ¯ bar(X)\bar{X}X¯ has a Mori fiber space structure X ¯ Z ¯ X ¯ → Z ¯ bar(X)rarr bar(Z)\bar{X} \rightarrow \bar{Z}X¯→Z¯ or K X ¯ K X ¯ K_( bar(X))K_{\bar{X}}KX¯ is nef. One of the remaining open problems is the termination of the program, to be more precise, termination of a sequence of flips. The problem was solved affirmatively in dimension 4 ≤ 4 <= 4\leq 4≤4 [35,69], for varieties of general type, for uniruled varieties [5], and in some other special cases. We refer to [3] for more comprehensive survey of the higher-dimensional MMP.
The MMP has a huge number of applications in algebraic geometry. The most impressive consequence of the MMP is the finite generation of the canonical ring
R ( X , K X ) := n 0 H 0 ( X , O X ( m K X ) ) R X , K X := ⨁ n ≥ 0   H 0 X , O X m K X R(X,K_(X)):=bigoplus_(n >= 0)H^(0)(X,O_(X)(mK_(X)))\mathrm{R}\left(X, K_{X}\right):=\bigoplus_{n \geq 0} H^{0}\left(X, \mathscr{O}_{X}\left(m K_{X}\right)\right)R(X,KX):=⨁n≥0H0(X,OX(mKX))
of a smooth projective variety X [ 5 , 15 ] X [ 5 , 15 ] X[5,15]X[5,15]X[5,15]. Another application of the MMP is the so-called Sarkisov program which allows decomposing birational maps between Mori fiber spaces into composition of elementary transformations, called Sarkisov links [9,16,68]. Also the MMP can be applied to varieties with finite group actions and to varieties over nonclosed fields (see [63]).
As was explained above, the Mori contractions are fundamental building blocks in the MMP. To apply the MMP effectively, one needs to understand the structure of its steps in details. For a Mori contraction f : X Z f : X → Z f:X rarr Zf: X \rightarrow Zf:X→Z of a three-dimensional variety X X XXX, there are only the following possibilities:
  • f f fff is divisorial and the image of the (prime) exceptional divisor E := Exc ( f ) E := Exc ⁡ ( f ) E:=Exc(f)E:=\operatorname{Exc}(f)E:=Exc⁡(f) is either a point or an irreducible curve,
  • f f fff is flipping and the exceptional locus Exc ( f ) Exc ⁡ ( f ) Exc(f)\operatorname{Exc}(f)Exc⁡(f) is a union of a finite number of irreducible curves,
  • Z Z ZZZ is a surface and f f fff is a Q Q Q\mathbb{Q}Q-conic bundle,
  • Z Z ZZZ is a curve and f f fff is a Q Q Q\mathbb{Q}Q-del Pezzo fibration,
  • Z Z ZZZ is a point and X X XXX is a Q Q Q\mathbb{Q}Q-Fano threefold.
Mori contractions of smooth threefolds to varieties of positive dimension where classified in the pioneering work of S. Mori [42]. S. Cutkosky [12] extended this classification to the case of Gorenstein terminal varieties. Smooth Fano threefolds of Picard number 1 where classified by Iskovskikh [22, 23] (see also [25]). Fano threefolds with Gorenstein terminal singularities are degenerated smooth ones [57]. Below we are going to discuss Mori contractions
of threefolds. We are interested only in the biregular structure of a contraction f : X Z f : X → Z f:X rarr Zf: X \rightarrow Zf:X→Z near a fixed fiber f 1 ( o ) , o Z f − 1 ( o ) , o ∈ Z f^(-1)(o),o in Zf^{-1}(o), o \in Zf−1(o),o∈Z. Typically, we do not consider the simple case where X X XXX is Gorenstein.

3. GENERAL ELEPHANT

A natural way to study higher-dimensional varieties is the inductive one. Typically, to apply this method, we need to find a certain subvariety of dimension one less (divisor) which is sufficiently good is the sense of singularities.
Conjecture 3.1. Let f : X ( Z o ) f : X → ( Z ∋ o ) f:X rarr(Z∋o)f: X \rightarrow(Z \ni o)f:X→(Z∋o) be a threefold Mori contraction, where ( Z o ) ( Z ∋ o ) (Z∋o)(Z \ni o)(Z∋o) is a small neighborhood. Then the general member D | K X | D ∈ − K X D in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| is a normal surface with Du Val singularities.
The conjecture was proposed by M. Reid who called a good member of | K X | − K X |-K_(X)|\left|-K_{X}\right||−KX| "elephant." We follow this language and call Conjecture 3.1 the General Elephant Conjecture. The importance of the existence of a good member in | K X | − K X |-K_(X)|\left|-K_{X}\right||−KX| is motivated by many reasons:
  • The general elephant passes through all the non-Gorenstein points of X X XXX and so it encodes the information about their types and configuration (cf. Proposition 3.2 below).
  • For flipping contractions, Conjecture 3.1 is a sufficient condition for the existence of threefold flips [32].
  • For a divisorial contraction f : X Z f : X → Z f:X rarr Zf: X \rightarrow Zf:X→Z whose fibers have dimension 1 ≤ 1 <= 1\leq 1≤1, the image D Z := f ( D ) D Z := f ( D ) D_(Z):=f(D)D_{Z}:=f(D)DZ:=f(D) of a Du Val elephant D | K X | D ∈ − K X D in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| must be again Du Val and the image Γ := f ( E ) Γ := f ( E ) Gamma:=f(E)\Gamma:=f(E)Γ:=f(E) of the exceptional divisor is a curve on D Z D Z D_(Z)D_{Z}DZ. Then one can reconstruct f f fff starting from the triple ( Z D Z Γ ) Z ⊃ D Z ⊃ Γ (Z supD_(Z)sup Gamma)\left(Z \supset D_{Z} \supset \Gamma\right)(Z⊃DZ⊃Γ) by using a certain birational procedure. Such an approach was successfully worked out in many cases by N. Tziolas [71-74].
  • If f : X ( Z o ) f : X → ( Z ∋ o ) f:X rarr(Z∋o)f: X \rightarrow(Z \ni o)f:X→(Z∋o) is a Q Q Q\mathbb{Q}Q-del Pezzo fibration such that general D | K X | D ∈ − K X D in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| is Du Val, then, composing the projection D Z D → Z D rarr ZD \rightarrow ZD→Z with minimal resolution D ~ D D ~ → D tilde(D)rarr D\tilde{D} \rightarrow DD~→D, we obtain a relatively minimal elliptic fibration whose singular fibers are classified by Kodaira [36]. Then one can get a bound of multiplicities of fibers and describe the configuration of non-Gorenstein singularities.
  • For a Q Q Q\mathbb{Q}Q-Fano threefold X X XXX, a Du Val general elephant is a (singular) K3 surface. In the case where the linear system | K X | − K X |-K_(X)|\left|-K_{X}\right||−KX| is "sufficiently big," this implies the existence of a good Gorenstein model [1].
Shokurov [70] generalized Conjecture 3.1 and introduced a new notion which is very efficient in the study of pluri-anticanonical linear systems. Omitting technicalities, we reproduce a weak form of Shokurov's definition.
Definition. An n n nnn-complement of the canonical class K X K X K_(X)K_{X}KX is a member D | n K X | D ∈ − n K X D in|-nK_(X)|D \in\left|-n K_{X}\right|D∈|−nKX| such that the pair ( X , 1 n D ) X , 1 n D (X,(1)/(n)D)\left(X, \frac{1}{n} D\right)(X,1nD) is lc. An n n nnn-complement is said to be klt (resp. plt) if such is the pair ( X , 1 n D ) X , 1 n D (X,(1)/(n)D)\left(X, \frac{1}{n} D\right)(X,1nD).
According to the inversion of adjunction [70], the existence of a Du Val general elephant D | K X | D ∈ − K X D in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| is equivalent to the existence of a plt 1-complement. Shokurov developed a powerful theory that works in arbitrary dimension and allows constructing complements inductively (see [ 64 , 70 ] [ 64 , 70 ] [64,70][64,70][64,70] and references therein).
Note that Reid's general elephant fails for Fano threefolds. For example, in [ 6 , 21 ] [ 6 , 21 ] [6,21][6,21][6,21] one can find examples of Q Q Q\mathbb{Q}Q-Fano threefolds with an empty anticanonical linear system. Because of this, the statement of Conjecture 3.1 sometimes is called a "principle." Nonetheless, there are only a few examples of such Fano threefolds. In the case dim ( Z ) > 0 dim ⁡ ( Z ) > 0 dim(Z) > 0\operatorname{dim}(Z)>0dim⁡(Z)>0, Conjecture 3.1 is expected to be true without exceptions. The following should be considered as the local version of Conjecture 3.1.
Proposition 3.2 (Reid [66]). Let ( X P ( X ∋ P (X∋P(X \ni P(X∋P ) be the analytic germ of a threefold terminal singularity of index m > 1 m > 1 m > 1m>1m>1. Then the general member D | K X | D ∈ − K X D in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| is a Du Val singularity. Furthermore, let π : X X Ï€ : X ′ → X pi:X^(')rarr X\pi: X^{\prime} \rightarrow XÏ€:X′→X be the index-one cover and let D := π 1 ( D ) D ′ := Ï€ − 1 ( D ) D^('):=pi^(-1)(D)D^{\prime}:=\pi^{-1}(D)D′:=π−1(D). Then the cover D D D ′ → D D^(')rarr DD^{\prime} \rightarrow DD′→D belongs to one of the following six types:
( X P ) ( X ∋ P ) (X∋P)(X \ni P)(X∋P) D D D ′ → D D^(')rarr DD^{\prime} \rightarrow DD′→D ( X P ) ( X ∋ P ) (X∋P)(X \ni P)(X∋P) D D D ′ → D D^(')rarr DD^{\prime} \rightarrow DD′→D
c A / m c A / m cA//m\mathrm{cA} / \mathrm{m}cA/m A k 1 m : 1 A k m 1 A k − 1 → m : 1 A k m − 1 A_(k-1)rarr"m:1"A_(km-1)\mathrm{A}_{\mathrm{k}-1} \xrightarrow{m: 1} \mathrm{~A}_{\mathrm{km}-1}Ak−1→m:1 Akm−1 c A x / 2 c A x / 2 cAx//2\mathrm{cAx} / 2cAx/2 A 2 k 1 2 : 1 D k + 2 A 2 k − 1 → 2 : 1 D k + 2 A_(2k-1)rarr"2:1"D_(k+2)\mathrm{~A}_{2 \mathrm{k}-1} \xrightarrow{2: 1} \mathrm{D}_{\mathrm{k}+2} A2k−1→2:1Dk+2
c A x / 4 c A x / 4 cAx//4\mathrm{cAx} / 4cAx/4 A 2 k 2 4 : 1 D 2 k + 1 A 2 k − 2 → 4 : 1 D 2 k + 1 A_(2k-2)rarr"4:1"D_(2k+1)\mathrm{~A}_{2 \mathrm{k}-2} \xrightarrow{4: 1} \mathrm{D}_{2 \mathrm{k}+1} A2k−2→4:1D2k+1 c D / 2 c D / 2 cD//2\mathrm{cD} / 2cD/2 D k + 1 2 : 1 D 2 k D k + 1 → 2 : 1 D 2 k D_(k+1)rarr"2:1"D_(2k)\mathrm{D}_{\mathrm{k}+1} \xrightarrow{2: 1} \mathrm{D}_{2 \mathrm{k}}Dk+1→2:1D2k
c D / 3 c D / 3 cD//3\mathrm{cD} / 3cD/3 D 4 3 : 1 E 6 D 4 → 3 : 1 E 6 D_(4)rarr"3:1"E_(6)\mathrm{D}_{4} \xrightarrow{3: 1} \mathrm{E}_{6}D4→3:1E6 c E / 2 c E / 2 cE//2\mathrm{cE} / 2cE/2 E 6 2 : 1 E 7 E 6 → 2 : 1 E 7 E_(6)rarr"2:1"E_(7)\mathrm{E}_{6} \xrightarrow{2: 1} \mathrm{E}_{7}E6→2:1E7
(X∋P) D^(')rarr D (X∋P) D^(')rarr D cA//m A_(k-1)rarr"m:1"A_(km-1) cAx//2 A_(2k-1)rarr"2:1"D_(k+2) cAx//4 A_(2k-2)rarr"4:1"D_(2k+1) cD//2 D_(k+1)rarr"2:1"D_(2k) cD//3 D_(4)rarr"3:1"E_(6) cE//2 E_(6)rarr"2:1"E_(7)| $(X \ni P)$ | $D^{\prime} \rightarrow D$ | $(X \ni P)$ | $D^{\prime} \rightarrow D$ | | :--- | :--- | :--- | :--- | | $\mathrm{cA} / \mathrm{m}$ | $\mathrm{A}_{\mathrm{k}-1} \xrightarrow{m: 1} \mathrm{~A}_{\mathrm{km}-1}$ | $\mathrm{cAx} / 2$ | $\mathrm{~A}_{2 \mathrm{k}-1} \xrightarrow{2: 1} \mathrm{D}_{\mathrm{k}+2}$ | | $\mathrm{cAx} / 4$ | $\mathrm{~A}_{2 \mathrm{k}-2} \xrightarrow{4: 1} \mathrm{D}_{2 \mathrm{k}+1}$ | $\mathrm{cD} / 2$ | $\mathrm{D}_{\mathrm{k}+1} \xrightarrow{2: 1} \mathrm{D}_{2 \mathrm{k}}$ | | $\mathrm{cD} / 3$ | $\mathrm{D}_{4} \xrightarrow{3: 1} \mathrm{E}_{6}$ | $\mathrm{cE} / 2$ | $\mathrm{E}_{6} \xrightarrow{2: 1} \mathrm{E}_{7}$ |

4. DIVISORIAL CONTRACTIONS TO A POINT

In this section we treat divisorial Mori contractions of a divisor to a point. Such contractions are studied very well due to works of Y. Kawamata [34], A. Corti [10], M. Kawakita [26-30], T. Hayakawa [18-20], and others. In this case, General Elephant Conjecture 3.1 has been verified:
Theorem 4.1 (Kawakita [28,29]). Let f : X ( Z o ) f : X → ( Z ∋ o ) f:X rarr(Z∋o)f: X \rightarrow(Z \ni o)f:X→(Z∋o) be a divisorial Mori contraction that contracts a divisor to a point. Then the general member D | K X | D ∈ − K X D in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| is Du Val.
One of the main tools in the proofs is the orbifold Riemann-Roch formula [66]: if X X XXX is a three-dimensional projective variety with terminal singularities and D D DDD is a Weil Q Q Q\mathbb{Q}Q-Cartier divisor on X X XXX, then for the sheaf L = O X ( D ) L = O X ( D ) L=O_(X)(D)\mathscr{L}=\mathscr{O}_{X}(D)L=OX(D) there is a formula of the form
(4.1.1) χ ( L ) = χ ( O X ) + 1 12 D ( D K X ) ( 2 D K X ) + 1 12 D c 2 + P c P ( D ) (4.1.1) χ ( L ) = χ O X + 1 12 D â‹… D − K X â‹… 2 D − K X + 1 12 D â‹… c 2 + ∑ P   c P ( D ) {:(4.1.1)chi(L)=chi(O_(X))+(1)/(12)D*(D-K_(X))*(2D-K_(X))+(1)/(12)D*c_(2)+sum_(P)c_(P)(D):}\begin{equation*} \chi(\mathscr{L})=\chi\left(\mathscr{O}_{X}\right)+\frac{1}{12} D \cdot\left(D-K_{X}\right) \cdot\left(2 D-K_{X}\right)+\frac{1}{12} D \cdot \mathrm{c}_{2}+\sum_{P} c_{P}(D) \tag{4.1.1} \end{equation*}(4.1.1)χ(L)=χ(OX)+112Dâ‹…(D−KX)â‹…(2D−KX)+112Dâ‹…c2+∑PcP(D)
where the sum rungs over all the virtual quotient singularities of X X XXX, i.e., over the actual singularities of X X XXX that are replaced with their small deformations [66], and c P ( D ) c P ( D ) c_(P)(D)c_{P}(D)cP(D) is a local
contribution due to singularity at P P PPP, depending only on the local analytic type of D D DDD at P P PPP. There is an explicit formula for the computation of c P ( D ) c P ( D ) c_(P)(D)c_{P}(D)cP(D).
Except for a few hard cases, the classification of divisorial Mori contractions of a divisor to a point has been completed. A typical result here is to show that a contraction is a weighted blowup with some explicit collection of weights:
Theorem 4.2 (Y. Kawamata [34]). Let f : X ( Z o ) f : X → ( Z ∋ o ) f:X rarr(Z∋o)f: X \rightarrow(Z \ni o)f:X→(Z∋o) be a divisorial Mori contraction that contracts a divisor to a point. Assume that o Z o ∈ Z o in Zo \in Zo∈Z is a cyclic quotient singularity of type 1 r ( a , a , 1 ) 1 r ( a , − a , 1 ) (1)/(r)(a,-a,1)\frac{1}{r}(a,-a, 1)1r(a,−a,1). Then f f fff is the weighted blowup with weights ( a / r , 1 a / r , 1 / r ) ( a / r , 1 − a / r , 1 / r ) (a//r,1-a//r,1//r)(a / r, 1-a / r, 1 / r)(a/r,1−a/r,1/r).
Theorem 4.3 (M. Kawakita [26]). Let f : X ( Z o ) f : X → ( Z ∋ o ) f:X rarr(Z∋o)f: X \rightarrow(Z \ni o)f:X→(Z∋o) be a divisorial Mori contraction that contracts a divisor to a smooth point. Then f f fff is the weighted blowup with weights ( 1 , a , b ) ( 1 , a , b ) (1,a,b)(1, a, b)(1,a,b), where gcd ( a , b ) = 1 gcd ⁡ ( a , b ) = 1 gcd(a,b)=1\operatorname{gcd}(a, b)=1gcd⁡(a,b)=1.
These results are intensively used in the three-dimensional birational geometry, for example, in the proof of birational rigidity of index-1 Fano threefold weighted hypersurfaces [ 11 ] [ 11 ] [11][11][11].

5. DEL PEZZO FIBRATIONS

Much less is known about the local structure of Q Q Q\mathbb{Q}Q-del Pezzo fibrations. As was explained in Section 3, the existence of a Du Val general elephant would be very helpful in the study such contractions. However, in this case Conjecture 3.1 is established only in some special situations.
An important question that can be asked in the Del Pezzo fibration case is the presence of multiple fibers.
Theorem 5.1 ([49]). Let f : X Z f : X → Z f:X rarr Zf: X \rightarrow Zf:X→Z be a Q Q Q\mathbb{Q}Q-del Pezzo fibration and let f ( o ) = m o F o f ∗ ( o ) = m o F o f^(**)(o)=m_(o)F_(o)f^{*}(o)=m_{o} F_{o}f∗(o)=moFo be a special fiber of multiplicity m o m o m_(o)m_{o}mo. Then m o 6 m o ≤ 6 m_(o) <= 6m_{o} \leq 6mo≤6 and all the cases 1 m o 6 1 ≤ m o ≤ 6 1 <= m_(o) <= 61 \leq m_{o} \leq 61≤mo≤6 occur. Moreover, the possibilities for the local behavior of F o F o F_(o)F_{o}Fo near singular points are described.
The main idea of the proof is to apply the orbifold Riemann-Roch formula (4.1.1) to the divisor F o F o F_(o)F_{o}Fo and its multiples.
Example. Suppose that the cyclic group μ 4 μ 4 mu_(4)\mu_{4}μ4 acts on P x 1 × P y 1 × C t P x 1 × P y 1 × C t P_(x)^(1)xxP_(y)^(1)xxC_(t)\mathbb{P}_{x}^{1} \times \mathbb{P}_{y}^{1} \times \mathbb{C}_{t}Px1×Py1×Ct via
( x , y ; t ) ( y , x , 1 t ) ( x , y ; t ) ⟼ ( y , − x , − 1 t ) (x,y;t)longmapsto(y,-x,sqrt(-1)t)(x, y ; t) \longmapsto(y,-x, \sqrt{-1} t)(x,y;t)⟼(y,−x,−1t)
Then the quotient
X = ( P 1 × P 1 × C ) / μ 4 Z = C / μ 4 X = P 1 × P 1 × C / μ 4 ⟶ Z = C / μ 4 X=(P^(1)xxP^(1)xxC)//mu_(4)longrightarrow Z=C//mu_(4)X=\left(\mathbb{P}^{1} \times \mathbb{P}^{1} \times \mathbb{C}\right) / \mu_{4} \longrightarrow Z=\mathbb{C} / \mu_{4}X=(P1×P1×C)/μ4⟶Z=C/μ4
is the germ of a Q Q Q\mathbb{Q}Q-del Pezzo fibration with central fiber of multiplicity 4.
Another type of Q Q Q\mathbb{Q}Q-del Pezzo fibrations which are investigated relatively well are those whose central fiber F := f 1 ( o ) F := f − 1 ( o ) F:=f^(-1)(o)F:=f^{-1}(o)F:=f−1(o) is reduced, normal, and has "good" singularities. Then X X XXX can be viewed as a one-parameter smoothing of F F FFF. The total space of this smoothing must be Q Q Q\mathbb{Q}Q-Gorenstein and F F FFF can be viewed as a degeneration of a general fiber (smooth del Pezzo surface) in a Q Q Q\mathbb{Q}Q-Gorenstein family. The most studied class of singularities admitting Q Q Q\mathbb{Q}Q-Gorenstein smoothings is the class of singularities of type T.
Definition (Kollár, Shepherd-Barron [40]). A two-dimensional quotient singularity is said to be of type T T T\mathrm{T}T if it admits a smoothing in a one-parameter Q Q Q\mathbb{Q}Q-Gorenstein family X B X → B X rarr BX \rightarrow BX→B.
In this case, by the inversion of adjunction [70], the pair ( X , F ) ( X , F ) (X,F)(X, F)(X,F) is plt and the total family X X XXX is terminal. Conversely, if X P X ∋ P X∋PX \ni PX∋P is a Q Q Q\mathbb{Q}Q-Gorenstein point and F F FFF is an effective Cartier divisor at P P PPP such that the pair ( X , F ) ( X , F ) (X,F)(X, F)(X,F) is plt, then F P F ∋ P F∋PF \ni PF∋P is a T-singularity and the point X P X ∋ P X∋PX \ni PX∋P is terminal. Singularities of type T and their deformations were studied by Kollár and Shepherd-Barron [40]. In particular, they proved that any T-singularity is either a Du Val point or a cyclic quotient of type 1 m ( q 1 , q 2 ) 1 m q 1 , q 2 (1)/(m)(q_(1),q_(2))\frac{1}{m}\left(q_{1}, q_{2}\right)1m(q1,q2) with
gcd ( m , q 1 ) = gcd ( m , q 2 ) = 1 , ( q 1 + q 2 ) 2 0 mod m gcd ⁡ m , q 1 = gcd ⁡ m , q 2 = 1 , q 1 + q 2 2 ≡ 0 mod m gcd(m,q_(1))=gcd(m,q_(2))=1,quad(q_(1)+q_(2))^(2)-=0mod m\operatorname{gcd}\left(m, q_{1}\right)=\operatorname{gcd}\left(m, q_{2}\right)=1, \quad\left(q_{1}+q_{2}\right)^{2} \equiv 0 \bmod mgcd⁡(m,q1)=gcd⁡(m,q2)=1,(q1+q2)2≡0modm
Minimal resolutions of these singularities are also described [40, § 3].
Thus to study Q Q Q\mathbb{Q}Q-del Pezzo fibrations whose central fiber has only quotient singularities, one has to consider Q Q Q\mathbb{Q}Q-Gorenstein smoothings of del Pezzo surfaces with singularities of type T. The important auxiliary fact here is the unobstructedness of deformations:
Proposition 5.2 ( [ 13 , 41 ] ) [ 13 , 41 ] ) [13,41])[13,41])[13,41]). Let F F FFF be a projective surface with log canonical singularities such that K F − K F -K_(F)-K_{F}−KF is big. Then there are no local-to-global obstructions to deformations of F F FFF. In particular, if F F FFF has T T T\mathrm{T}T-singularities, then F F FFF admits a Q Q Q\mathbb{Q}Q-Gorenstein smoothing.
Theorem 5.3 (Hacking-Prokhorov [13]). Let F be a projective surface with quotient singularities such that K F − K F -K_(F)-K_{F}−KF is ample, ρ ( F ) = 1 ρ ( F ) = 1 rho(F)=1\rho(F)=1ρ(F)=1, and F F FFF admits a Q Q Q\mathbb{Q}Q-Gorenstein smoothing. Then F F FFF belongs to one of the following:
  • 14 infinite sequences of toric surfaces (see below);
  • partial smoothing of a toric surface as above;
  • 18 sporadic families of surfaces of index 2 ≤ 2 <= 2\leq 2≤2 [2].
Toric surfaces appearing in the above theorem are determined by a Markov-type equation. More precisely, for K F 2 5 K F 2 ≥ 5 K_(F)^(2) >= 5K_{F}^{2} \geq 5KF2≥5 these surfaces are weighted projective spaces given by the following table:
K F 2 K F 2 K_(F)^(2)K_{F}^{2}KF2 F F FFF Markov-type equation
9 P ( a 2 , b 2 , c 2 ) P a 2 , b 2 , c 2 P(a^(2),b^(2),c^(2))\mathbb{P}\left(a^{2}, b^{2}, c^{2}\right)P(a2,b2,c2) a 2 + b 2 + c 2 = 3 a b c a 2 + b 2 + c 2 = 3 a b c a^(2)+b^(2)+c^(2)=3abca^{2}+b^{2}+c^{2}=3 a b ca2+b2+c2=3abc
8 P ( a 2 , b 2 , 2 c 2 ) P a 2 , b 2 , 2 c 2 P(a^(2),b^(2),2c^(2))\mathbb{P}\left(a^{2}, b^{2}, 2 c^{2}\right)P(a2,b2,2c2) a 2 + b 2 + 2 c 2 = 4 a b c a 2 + b 2 + 2 c 2 = 4 a b c a^(2)+b^(2)+2c^(2)=4abca^{2}+b^{2}+2 c^{2}=4 a b ca2+b2+2c2=4abc
6 P ( a 2 , 2 b 2 , 3 c 2 ) P a 2 , 2 b 2 , 3 c 2 P(a^(2),2b^(2),3c^(2))\mathbb{P}\left(a^{2}, 2 b^{2}, 3 c^{2}\right)P(a2,2b2,3c2) a 2 + 2 b 2 + 3 c 2 = 6 a b c a 2 + 2 b 2 + 3 c 2 = 6 a b c a^(2)+2b^(2)+3c^(2)=6abca^{2}+2 b^{2}+3 c^{2}=6 a b ca2+2b2+3c2=6abc
5 P ( a 2 , b 2 , 5 c 2 ) P a 2 , b 2 , 5 c 2 P(a^(2),b^(2),5c^(2))\mathbb{P}\left(a^{2}, b^{2}, 5 c^{2}\right)P(a2,b2,5c2) a 2 + b 2 + 5 c 2 = 5 a b c a 2 + b 2 + 5 c 2 = 5 a b c a^(2)+b^(2)+5c^(2)=5abca^{2}+b^{2}+5 c^{2}=5 a b ca2+b2+5c2=5abc
K_(F)^(2) F Markov-type equation 9 P(a^(2),b^(2),c^(2)) a^(2)+b^(2)+c^(2)=3abc 8 P(a^(2),b^(2),2c^(2)) a^(2)+b^(2)+2c^(2)=4abc 6 P(a^(2),2b^(2),3c^(2)) a^(2)+2b^(2)+3c^(2)=6abc 5 P(a^(2),b^(2),5c^(2)) a^(2)+b^(2)+5c^(2)=5abc| $K_{F}^{2}$ | $F$ | Markov-type equation | | ---: | :--- | :--- | | 9 | $\mathbb{P}\left(a^{2}, b^{2}, c^{2}\right)$ | $a^{2}+b^{2}+c^{2}=3 a b c$ | | 8 | $\mathbb{P}\left(a^{2}, b^{2}, 2 c^{2}\right)$ | $a^{2}+b^{2}+2 c^{2}=4 a b c$ | | 6 | $\mathbb{P}\left(a^{2}, 2 b^{2}, 3 c^{2}\right)$ | $a^{2}+2 b^{2}+3 c^{2}=6 a b c$ | | 5 | $\mathbb{P}\left(a^{2}, b^{2}, 5 c^{2}\right)$ | $a^{2}+b^{2}+5 c^{2}=5 a b c$ |
and for K 2 4 K 2 ≤ 4 K^(2) <= 4K^{2} \leq 4K2≤4 they are certain abelian quotients of the weighted projective spaces as above. Note, however, that in general we cannot assert that, for central fiber F F FFF of a Q Q Q\mathbb{Q}Q-del Pezzo fibration, the condition ρ ( F ) = 1 ρ ( F ) = 1 rho(F)=1\rho(F)=1ρ(F)=1 holds. Some partial results in the case ρ ( F ) > 1 ρ ( F ) > 1 rho(F) > 1\rho(F)>1ρ(F)>1 where obtained in [60]. In particular, [60] establishes the existence of Du Val general elephant for Q Q Q\mathbb{Q}Q-del Pezzo fibrations with "good" fibers:
Theorem 5.4. Let f : X ( Z o ) f : X → ( Z ∋ o ) f:X rarr(Z∋o)f: X \rightarrow(Z \ni o)f:X→(Z∋o) be a Q Q Q\mathbb{Q}Q-del Pezzo fibration over a curve germ Z o Z ∋ o Z∋oZ \ni oZ∋o. Assume that the fiber f 1 ( o ) f − 1 ( o ) f^(-1)(o)f^{-1}(o)f−1(o) is reduced, normal, and has only log terminal singularities. Then the general elephant D | K X | D ∈ − K X D in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| is Du Val.
Theorem 5.3 gives a complete answer to the question posed by M. Manetti [41]:
Corollary 5.5 ([13]). Let X X XXX be a projective surface with quotient singularities which admits a smoothing to P 2 P 2 P^(2)\mathbb{P}^{2}P2. Then X X XXX is a Q Q Q\mathbb{Q}Q-Gorenstein deformation of a weighted projective plane P ( a 2 , b 2 , c 2 ) P a 2 , b 2 , c 2 P(a^(2),b^(2),c^(2))\mathbb{P}\left(a^{2}, b^{2}, c^{2}\right)P(a2,b2,c2), where the triple ( a , b , c ) ( a , b , c ) (a,b,c)(a, b, c)(a,b,c) is a solution of the Markov equation
a 2 + b 2 + c 2 = 3 a b c a 2 + b 2 + c 2 = 3 a b c a^(2)+b^(2)+c^(2)=3abca^{2}+b^{2}+c^{2}=3 a b ca2+b2+c2=3abc
Results similar to Theorem 5.3 were obtained for Q Q Q\mathbb{Q}Q-del Pezzo fibrations whose central fiber is log canonical [62]. However, in this case the classification is not complete.

6. EXTREMAL CURVE GERMS

To study Mori contractions with fibers of dimension 1 ≤ 1 <= 1\leq 1≤1, it is convenient to work with analytic threefolds and to localize to situation near a curve contained in a fiber.
Definition 6.1. Let ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) be the analytic germ of a threefold with terminal singularities along a reduced connected complete curve. Then ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) is called an extremal curve germ if there exists a contraction
f : ( X C ) ( Z o ) f : ( X ⊃ C ) ⟶ ( Z ∋ o ) f:(X sup C)longrightarrow(Z∋o)f:(X \supset C) \longrightarrow(Z \ni o)f:(X⊃C)⟶(Z∋o)
such that C = f 1 ( o ) red C = f − 1 ( o ) red  C=f^(-1)(o)_("red ")C=f^{-1}(o)_{\text {red }}C=f−1(o)red  and K X − K X -K_(X)-K_{X}−KX is f f fff-ample. The curve C C CCC is called the central fiber of the germ and Z o Z ∋ o Z∋oZ \ni oZ∋o is called the target variety or the base of ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C). An extremal curve germ is said to be irreducible if such is its central fiber.
In the definition above, we do not assume that X X XXX is Q Q Q\mathbb{Q}Q-factorial or ρ ( X / Z ) = 1 ρ ( X / Z ) = 1 rho(X//Z)=1\rho(X / Z)=1ρ(X/Z)=1. This is because Q Q Q\mathbb{Q}Q-factoriality typically is not a local condition in the analytic category (see [ 32 , $ 1 ] ) [ 32 , $ 1 ] ) [32,$1])[32, \$ 1])[32,$1]). There are three types of extremal curve germs:
  • flipping if f f fff is birational and does not contract divisors;
  • divisorial if the exceptional locus is two-dimensional;
  • Q Q Q\mathbb{Q}Q-conic bundle germ if the target variety Z Z ZZZ is a surface.
If a divisorial curve germ is irreducible, then the exceptional locus of the corresponding contraction is a Q Q Q\mathbb{Q}Q-Cartier divisor and the target variety Z Z ZZZ has terminal singularities [51, §3]. In general, this is not true. It may happen that the exceptional locus is a union of a divisor and some curves.
As an example, we consider the case where X X XXX has singularities of indices 1 and 2 .
Theorem 6.2 ([47]). Let ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) be a Q Q Q\mathbb{Q}Q-conic bundle germ over a smooth base. Assume that X X XXX is not Gorenstein and 2 K X 2 K X 2K_(X)2 K_{X}2KX is Cartier. Then X X XXX can be embedded to P ( 1 , 1 , 1 , 2 ) × C 2 P ( 1 , 1 , 1 , 2 ) × C 2 P(1,1,1,2)xxC^(2)\mathbb{P}(1,1,1,2) \times \mathbb{C}^{2}P(1,1,1,2)×C2
and given there by two quadratic equations. In particular, the point P X P ∈ X P in XP \in XP∈X of index 2 is unique, the curve C C CCC has at most 4 components, all of them pass through P P PPP.
Theorem 6.3 ([38]). Let ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) be a flipping extremal curve germ and let
be the corresponding flip. Assume that 2 K X 2 K X 2K_(X)2 K_{X}2KX is Cartier. Then ( Z o ) ( Z ∋ o ) (Z∋o)(Z \ni o)(Z∋o) is the quotient of the isolated hypersurface singularity
{ x 1 x 3 + x 2 ϕ ( x 2 2 , x 4 ) = 0 } 0 x 1 x 3 + x 2 Ï• x 2 2 , x 4 = 0 ∋ 0 {x_(1)x_(3)+x_(2)phi(x_(2)^(2),x_(4))=0}∋0\left\{x_{1} x_{3}+x_{2} \phi\left(x_{2}^{2}, x_{4}\right)=0\right\} \ni 0{x1x3+x2Ï•(x22,x4)=0}∋0
by the μ 2 μ 2 mu_(2)\boldsymbol{\mu}_{2}μ2-action given by the weights ( 1 , 1 , 0 , 0 ) ( 1 , 1 , 0 , 0 ) (1,1,0,0)(1,1,0,0)(1,1,0,0). The contraction f f fff (resp. f + f + f^(+)f^{+}f+) is the quotient of the blowup of the plane { x 2 = x 3 = 0 } x 2 = x 3 = 0 {x_(2)=x_(3)=0}\left\{x_{2}=x_{3}=0\right\}{x2=x3=0} (resp. { x 1 = x 2 = 0 } x 1 = x 2 = 0 {x_(1)=x_(2)=0}\left\{x_{1}=x_{2}=0\right\}{x1=x2=0} ) by μ 2 μ 2 mu_(2)\mu_{2}μ2. In particular, X X XXX contains a unique point of index 2 and the central fiber C C CCC is irreducible. The variety X + X + X^(+)X^{+}X+ is Gorenstein.
A similar description is known for divisorial extremal curve germs of index 2 [ 38 , 84 ] [ 38 , 84 ] [38,84][38,84][38,84].
First properties. Let ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) be an extremal curve germ and let f : ( X C ) ( Z o ) f : ( X ⊃ C ) → ( Z ∋ o ) f:(X sup C)rarr(Z∋o)f:(X \supset C) \rightarrow(Z \ni o)f:(X⊃C)→(Z∋o) be the corresponding contraction. For any connected subcurve C C C ′ ⊂ C C^(')sub CC^{\prime} \subset CC′⊂C, the germ ( X C ) X ⊃ C ′ (X supC^('))\left(X \supset C^{\prime}\right)(X⊃C′) is again an extremal curve germ. If, moreover, C C C ′ ⫋ C C^(')⫋CC^{\prime} \varsubsetneqq CC′⫋C, then ( X C ) X ⊃ C ′ (X supC^('))\left(X \supset C^{\prime}\right)(X⊃C′) is birational. By the Kawamata-Viehweg vanishing theorem,
(6.3.1) R 1 f O X = 0 (6.3.1) R 1 f ∗ O X = 0 {:(6.3.1)R^(1)f_(**)O_(X)=0:}\begin{equation*} R^{1} f_{*} \mathscr{O}_{X}=0 \tag{6.3.1} \end{equation*}(6.3.1)R1f∗OX=0
(see, e.g., [35]). As a consequence, one has p a ( C ) 0 p a C ′ ≤ 0 p_(a)(C^(')) <= 0\mathrm{p}_{\mathrm{a}}\left(C^{\prime}\right) \leq 0pa(C′)≤0 for any subcurve C C C ′ ⊂ C C^(')sub CC^{\prime} \subset CC′⊂C. In particular, C = C i C = ⋃ C i C=uuuC_(i)C=\bigcup C_{i}C=⋃Ci is a "tree" of smooth rational curves. Furthermore,
(6.3.2) Pic ( X ) H 2 ( X , Z ) Z n (6.3.2) Pic ⁡ ( X ) ≃ H 2 ( X , Z ) ≃ Z ⊕ n {:(6.3.2)Pic(X)≃H^(2)(X","Z)≃Z^(o+n):}\begin{equation*} \operatorname{Pic}(X) \simeq H^{2}(X, \mathbb{Z}) \simeq \mathbb{Z}^{\oplus n} \tag{6.3.2} \end{equation*}(6.3.2)Pic⁡(X)≃H2(X,Z)≃Z⊕n
where n n nnn is the number of irreducible components of C C CCC. For more delicate properties of extremal curve germs, one needs to know the cohomology of the dualizing sheaf, see [44,47]:
(6.3.3) R 1 f ω X = { 0 , if f is birational, ω Z , if f is Q -conic bundle and Z is smooth. (6.3.3) R 1 f ∗ ω X = 0 ,  if  f  is birational,  ω Z ,  if  f  is  Q -conic bundle and  Z  is smooth.  {:(6.3.3)R^(1)f_(**)omega_(X)={[0","," if "f" is birational, "],[omega_(Z)","," if "f" is "Q"-conic bundle and "Z" is smooth. "]:}:}R^{1} f_{*} \omega_{X}= \begin{cases}0, & \text { if } f \text { is birational, } \tag{6.3.3}\\ \omega_{Z}, & \text { if } f \text { is } \mathbb{Q} \text {-conic bundle and } Z \text { is smooth. }\end{cases}(6.3.3)R1f∗ωX={0, if f is birational, ωZ, if f is Q-conic bundle and Z is smooth. 
Definition. An irreducible extremal curve germ ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) is (locally) imprimitive at a point P P PPP if the inverse image of C C CCC under the index-one cover ( X P ) ( X P ) X ♯ ∋ P ♯ → ( X ∋ P ) (X^(♯)∋P^(♯))rarr(X∋P)\left(X^{\sharp} \ni P^{\sharp}\right) \rightarrow(X \ni P)(X♯∋P♯)→(X∋P) splits.
Theorem 6.4 ([44,47]). Let ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) be an extremal curve germ and let C 1 , , C n C 1 , … , C n C_(1),dots,C_(n)C_{1}, \ldots, C_{n}C1,…,Cn be irreducible components of C C CCC.
  • Each C i C i C_(i)C_{i}Ci contains at most 3 singular points of X X XXX.
  • Each C i C i C_(i)C_{i}Ci contains at most 2 non-Gorenstein points of X X XXX and at most 1 point which is imprimitive for ( X C i ) X ⊃ C i (X supC_(i))\left(X \supset C_{i}\right)(X⊃Ci).
  • If X X XXX is Gorenstein at the intersection point P = C i C j , C i C j P = C i ∩ C j , C i ≠ C j P=C_(i)nnC_(j),C_(i)!=C_(j)P=C_{i} \cap C_{j}, C_{i} \neq C_{j}P=Ci∩Cj,Ci≠Cj, then X X XXX is smooth outside P P PPP and ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) is a Q Q Q\mathbb{Q}Q-conic bundle germ over a smooth base.
To prove the first assertion, one needs to analyze the conormal sheaf I C / I C 2 I C / I C 2 I_(C)//I_(C)^(2)I_{C} / I_{C}^{2}IC/IC2 and use the vanishing H 1 ( O X / J ) = 0 H 1 O X / J = 0 H^(1)(O_(X)//J)=0H^{1}\left(\mathscr{O}_{X} / J\right)=0H1(OX/J)=0 for any J O X J ⊂ O X J subO_(X)J \subset \mathscr{O}_{X}J⊂OX with Supp ( O X / J ) = C Supp ⁡ O X / J = C Supp(O_(X)//J)=C\operatorname{Supp}\left(\mathscr{O}_{X} / J\right)=CSupp⁡(OX/J)=C (see [44,55]). For the second assertion, one can use topological arguments based on (1.2.1) (see [55]). For the last assertion, we refer to [44, 1.15], [37, 4.2], and [55, 4.7.6]
The techniques applied in the proof of the above proposition allow obtaining much stronger results. In particular, they alow classifying all the possibilities for the local behavior of an irreducible germ ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) near a singular point P P PPP [44]. Thus, according to [44] and [47], the triple ( X C P ( X ⊃ C ∋ P (X sup C∋P(X \supset C \ni P(X⊃C∋P ) belongs to one of the following types:
(IA), (IC), (IIA), (IIB), (IA ) , ( I I ) , ( I D ) , ( I E ) , ( I I I )  (IA), (IC), (IIA), (IIB), (IA  ∨ , I I ∨ , I D ∨ , I E ∨ , ( I I I ) " (IA), (IC), (IIA), (IIB), (IA "{:^(vv)),(II^(vv)),(ID^(vv)),(IE^(vv)),(III)\text { (IA), (IC), (IIA), (IIB), (IA } \left.{ }^{\vee}\right),\left(\mathrm{II}^{\vee}\right),\left(\mathrm{ID}^{\vee}\right),\left(\mathrm{IE}^{\vee}\right),(\mathrm{III}) (IA), (IC), (IIA), (IIB), (IA ∨),(II∨),(ID∨),(IE∨),(III)
Here the symbol ∨ ^(vv){ }^{\vee}∨ means that ( X C P ) ( X ⊃ C ∋ P ) (X sup C∋P)(X \supset C \ni P)(X⊃C∋P) is locally imprimitive, the symbol II means that ( X P X ∋ P X∋PX \ni PX∋P ) is a terminal point of exceptional type cAx/4 (see Proposition 3.2), and III means that ( X P ( X ∋ P (X∋P(X \ni P(X∋P ) is an (isolated) cDV-point.
For example, a triple ( X C P X ⊃ C ∋ P X sup C∋PX \supset C \ni PX⊃C∋P ) is of type (IC) if there are analytic isomorphisms
( X P ) C y 1 , y 2 , y 4 3 / μ m ( 2 , m 2 , 1 ) , C { y 1 m 2 y 2 2 = y 4 = 0 } / μ m ( X ∋ P ) ≃ C y 1 , y 2 , y 4 3 / μ m ( 2 , m − 2 , 1 ) , C ≃ y 1 m − 2 − y 2 2 = y 4 = 0 / μ m (X∋P)≃C_(y_(1),y_(2),y_(4))^(3)//mu_(m)(2,m-2,1),quad C≃{y_(1)^(m-2)-y_(2)^(2)=y_(4)=0}//mu_(m)(X \ni P) \simeq \mathbb{C}_{y_{1}, y_{2}, y_{4}}^{3} / \mu_{m}(2, m-2,1), \quad C \simeq\left\{y_{1}^{m-2}-y_{2}^{2}=y_{4}=0\right\} / \mu_{m}(X∋P)≃Cy1,y2,y43/μm(2,m−2,1),C≃{y1m−2−y22=y4=0}/μm
where m m mmm is odd and m 5 m ≥ 5 m >= 5m \geq 5m≥5. For definitions other types, we refer to [44] and [47].

6.1. Construction of germs by deformations

Let ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) be an extremal curve germ and let f : X ( Z o ) f : X → ( Z ∋ o ) f:X rarr(Z∋o)f: X \rightarrow(Z \ni o)f:X→(Z∋o) be the corresponding contraction. Denote by | O Z | O Z |O_(Z)|\left|\mathscr{O}_{Z}\right||OZ| the infinite-dimensional linear system of hyperplane sections passing through o o ooo and let | O X | := f | O Z | O X := f ∗ O Z |O_(X)|:=f^(**)|O_(Z)|\left|\mathscr{O}_{X}\right|:=f^{*}\left|\mathscr{O}_{Z}\right||OX|:=f∗|OZ|. The general hyperplane section of ( X C X ⊃ C (X sup C:}\left(X \supset C\right.(X⊃C ) is the general member H | O X | H ∈ O X H in|O_(X)|H \in\left|\mathscr{O}_{X}\right|H∈|OX|. The divisor H H HHH contains much more information on the total space than a general elephant D | K X | D ∈ − K X D in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX|. However, the singularities of H H HHH typically are more complicated, in particular, H H HHH can be nonnormal.
The variety X X XXX (resp. Z Z ZZZ ) can be viewed as the total space of a one-parameter deformation of H H HHH (resp. H Z := f ( H ) H Z := f ( H ) H_(Z):=f(H)H_{Z}:=f(H)HZ:=f(H) ). We are going to reverse this consideration.
Construction (see [38, § 11], [44, § 1B]). Suppose we are given a normal surface germ ( H C H ⊃ C (H sup C:}\left(H \supset C\right.(H⊃C ) along a proper curve C C CCC and a contraction f H : H H Z f H : H → H Z f_(H):H rarrH_(Z)f_{H}: H \rightarrow H_{Z}fH:H→HZ such that C C CCC is a fiber and K H − K H -K_(H)-K_{H}−KH is f H f H f_(H)f_{H}fH-ample. Let P 1 , , P r H P 1 , … , P r ∈ H P_(1),dots,P_(r)in HP_{1}, \ldots, P_{r} \in HP1,…,Pr∈H be all the singular points. Assume also that near each P i P i P_(i)P_{i}Pi there exists a small one-parameter deformation S i S i S_(i)\mathfrak{S}_{i}Si of a neighborhood H i H i H_(i)H_{i}Hi of P i P i P_(i)P_{i}Pi in H H HHH such that the total space S i S i S_(i)\mathfrak{S}_{i}Si has a terminal singularity at P i P i P_(i)P_{i}Pi. The obstruction to globalize deformations
Ψ : Def ( H ) P i Sing ( H ) Def ( H , P i ) Ψ : Def ⁡ ( H ) ⟶ ∏ P i ∈ Sing ⁡ ( H )   Def ⁡ H , P i Psi:Def(H)longrightarrowprod_(P_(i)in Sing(H))Def(H,P_(i))\Psi: \operatorname{Def}(H) \longrightarrow \prod_{P_{i} \in \operatorname{Sing}(H)} \operatorname{Def}\left(H, P_{i}\right)Ψ:Def⁡(H)⟶∏Pi∈Sing⁡(H)Def⁡(H,Pi)
lies in R 2 f T H R 2 f ∗ T H R^(2)f_(**)T_(H)R^{2} f_{*} \mathcal{T}_{H}R2f∗TH, where T H = H T H = H T_(H)=H\mathcal{T}_{H}=\mathscr{H}TH=H om ( Ω H , O H ) Ω H , O H (Omega_(H),O_(H))\left(\Omega_{H}, \mathscr{O}_{H}\right)(ΩH,OH) is the tangent sheaf of H H HHH. Since R 2 f T H = 0 R 2 f ∗ T H = 0 R^(2)f_(**)T_(H)=0R^{2} f_{*} \mathcal{T}_{H}=0R2f∗TH=0 due to dimension reasons, the morphism Ψ Î¨ Psi\PsiΨ is smooth, and so there exists a global oneparameter deformation S S S\mathfrak{S}S of H H HHH inducing a local deformation of S i S i S_(i)\mathfrak{S}_{i}Si near P i P i P_(i)P_{i}Pi.
Then we have a threefold X := S C X := S ⊃ C X:=Ssup CX:=\mathfrak{S} \supset CX:=S⊃C with H | O X | H ∈ O X H in|O_(X)|H \in\left|\mathscr{O}_{X}\right|H∈|OX| such that locally near P i P i P_(i)P_{i}Pi it has the desired structure and one can extend f H f H f_(H)f_{H}fH to a contraction f : X Z f : X → Z f:X rarr Zf: X \rightarrow Zf:X→Z which is birational (resp. a Q Q Q\mathbb{Q}Q-conic bundle) if H Z H Z H_(Z)H_{Z}HZ is a surface (resp. a curve).
Example. Consider a rational curve fibration f H ~ : H ~ H Z f H ~ : H ~ → H Z f_( tilde(H)): tilde(H)rarrH_(Z)f_{\tilde{H}}: \tilde{H} \rightarrow H_{Z}fH~:H~→HZ over a smooth curve germ H Z o H Z ∋ o H_(Z)∋oH_{Z} \ni oHZ∋o, where H ~ H ~ tilde(H)\tilde{H}H~ is a smooth surface such that the fiber over o o ooo has the following weighted dual graph:
Contracting the curves corresponding to the white vertices ◻ ◻\square◻ and ∘ @\circ∘, we obtain a singular surface H H HHH and a K H K H K_(H)K_{H}KH-negative contraction f H : H H Z f H : H → H Z f_(H):H rarrH_(Z)f_{H}: H \rightarrow H_{Z}fH:H→HZ whose fiber over o o ooo is a curve C H C ⊂ H C sub HC \subset HC⊂H having three irreducible components that correspond to the black vertices ∙ ∙\bullet∙. The singular locus of H H HHH consists of a Du Val point P 0 H P 0 ∈ H P_(0)in HP_{0} \in HP0∈H of type A 1 A 1 A_(1)\mathrm{A}_{1}A1 and a log canonical singularity P H P ∈ H P in HP \in HP∈H whose dual graph is formed by the white circle vertices ∘ @\circ∘. Both P 0 P 0 P_(0)P_{0}P0 and P P PPP have 1-parameter Q Q Q\mathbb{Q}Q-Gorenstein smoothings [38, computation 6.7.1]. Applying the above construction to H C H ⊃ C H sup CH \supset CH⊃C, we obtain an example of a Q Q Q\mathbb{Q}Q-conic bundle contraction f : ( X C ) ( Z o ) f : ( X ⊃ C ) → ( Z ∋ o ) f:(X sup C)rarr(Z∋o)f:(X \supset C) \rightarrow(Z \ni o)f:(X⊃C)→(Z∋o) with a unique non-Gorenstein point which is of type cD/3. If we remove the (-2)-curve corresponding to ◻ ◻\square◻ on the left-hand side of the graph, we get a birational contraction of surfaces f H : H H Z f H ′ : H ′ → H Z ′ f_(H)^('):H^(')rarrH_(Z)^(')f_{H}^{\prime}: H^{\prime} \rightarrow H_{Z}^{\prime}fH′:H′→HZ′. Applying the same construction to H C H ′ ⊃ C H^(')sup CH^{\prime} \supset CH′⊃C, we obtain an example of a divisorial contraction. Similarly, removing further one of the ( 1 ) ( − 1 ) (-1)(-1)(−1)-curves, we get a flip.

7. EXTREMAL CURVE GERMS: GENERAL ELEPHANT

Theorem 7.1 (Mori [44], Kollár-Mori [38], Mori-Prokhorov [50]). Let ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) be an irreducible extremal curve germ. Then the general member D | K X | D ∈ − K X D in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| has only Du Val singularities.
The existence of a Du Val elephant for extremal curve germs with reducible central fiber is not known at the moment. See Theorem 9.2 below for partial results in this direction.
Comment on the proof. Essentially, there are three methods to find a good elephant D | K X | D ∈ − K X D in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX|. We outline them below.

7.1. Local method

As in Proposition 3.2, near each non-Gorenstein point P i X P i ∈ X P_(i)in XP_{i} \in XPi∈X take a local general elephant D i | K ( X P i ) | D i ∈ − K X ∋ P i D_(i)in|-K_((X∋P_(i)))|D_{i} \in\left|-K_{\left(X \ni P_{i}\right)}\right|Di∈|−K(X∋Pi)|. Since D i D i D_(i)D_{i}Di is general, we have D i C = { P i } D i ∩ C = P i D_(i)nn C={P_(i)}D_{i} \cap C=\left\{P_{i}\right\}Di∩C={Pi}. Then we can regard D := D i D := ∑ D i D:=sumD_(i)D:=\sum D_{i}D:=∑Di as a Weil divisor on X X XXX. By the construction, K X + D K X + D K_(X)+DK_{X}+DKX+D is a Cartier divisor near each P i P i P_(i)P_{i}Pi, hence it is Cartier everywhere. In some cases it is possible to compute the intersection numbers D i C D i ⋅ C D_(i)*CD_{i} \cdot CDi⋅C and show that D C < 1 D ⋅ C < 1 D*C < 1D \cdot C<1D⋅C<1. Then we may assume that K X + D 0 K X + D ∼ 0 K_(X)+D∼0K_{X}+D \sim 0KX+D∼0 by (6.3.2) and so D D DDD is a Du Val anticanonical divisor. For example, this method works for extremal curve germs described in Theorems 6.2 and 6.3, as well as in Example 7.3 below.

7.2. Extension from S | 2 K X | S ∈ − 2 K X S in|-2K_(X)|S \in\left|-2 K_{X}\right|S∈|−2KX|

In some cases, the above approach does not work, but it allows showing the existence of a klt 2-complement S | 2 K X | S ∈ − 2 K X S in|-2K_(X)|S \in\left|-2 K_{X}\right|S∈|−2KX| such that dim ( D C ) = 0 dim ⁡ ( D ∩ C ) = 0 dim(D nn C)=0\operatorname{dim}(D \cap C)=0dim⁡(D∩C)=0. Then one can try to extend a good element from the surface S S SSS. The crucial fact here is that the natural map
τ : H 0 ( X , O X ( K X ) ) H 0 ( S , O S ( K X ) ) = ω S Ï„ : H 0 X , O X − K X ⟶ H 0 S , O S − K X = ω S tau:H^(0)(X,O_(X)(-K_(X)))longrightarrowH^(0)(S,O_(S)(-K_(X)))=omega_(S)\tau: H^{0}\left(X, \mathscr{O}_{X}\left(-K_{X}\right)\right) \longrightarrow H^{0}\left(S, \mathscr{O}_{S}\left(-K_{X}\right)\right)=\omega_{S}Ï„:H0(X,OX(−KX))⟶H0(S,OS(−KX))=ωS
is surjective if ( X , C ) ( X , C ) (X,C)(X, C)(X,C) is birational and surjective modulo Ω S 2 Ω S 2 Omega_(S)^(2)\Omega_{S}^{2}ΩS2 if ( X , C ) ( X , C ) (X,C)(X, C)(X,C) is a Q Q Q\mathbb{Q}Q-conic bundle. This immediately follows from (6.3.3). Details can be found in [38, § 2] and [50].

7.3. Global method

Finally, in the most complicated cases, none of the above methods work. Then one needs more subtle techniques which require detailed analysis of singularities and infinitesimal structure of X X XXX along C C CCC [44, §§ 8-9]. Then, roughly speaking, the good section D | K X | D ∈ − K X D in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| is recovered as the formal Weil divisor lim C n lim C n limC_(n)\lim C_{n}limCn of the completion X X ∧ X^X^{\wedge}X∧ of X X XXX along C C CCC, where C n C n C_(n)C_{n}Cn are subschemes with support C C CCC constructed by using certain inductive procedure [ 44 , § 9 ] [ 44 , § 9 ] [44,§9][44, \S 9][44,§9].
As a consequence of Theorem 7.1, in the Q Q Q\mathbb{Q}Q-conic bundle case, one obtains the following fact which proves Iskovskikh's conjecture [24].
Corollary 7.2. Let ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) be a Q Q Q\mathbb{Q}Q-conic bundle germ over ( Z o ) ( Z ∋ o ) (Z∋o)(Z \ni o)(Z∋o), where C C CCC can be reducible. Then ( Z o ) ( Z ∋ o ) (Z∋o)(Z \ni o)(Z∋o) is a Du Val singularity of type A n A n A_(n)\mathrm{A}_{\mathrm{n}}An (or smooth).
This result is very useful for applications to the rationality problem of three-dimensional varieties having conic bundle structure [ 24 , 61 ] [ 24 , 61 ] [24,61][24,61][24,61] and some problems of biregular geometry [ 58 , 59 ] [ 58 , 59 ] [58,59][58,59][58,59].
It turns out that the structure of Q Q Q\mathbb{Q}Q-conic bundle germs over a singular base ( Z o Z ∋ o Z∋oZ \ni oZ∋o ) is much simpler and shorter than others. In fact, these germs can be exhibited as certain quotients of Q Q Q\mathbb{Q}Q-conic bundles of index 2 ≤ 2 <= 2\leq 2≤2 (see Theorem 6.2). A complete classification of such germs was obtained in [ 47 , 48 ] [ 47 , 48 ] [47,48][47,48][47,48]. Here is a typical example.
Example 7.3. Let the group μ n μ n mu_(n)\mu_{n}μn act on C u , v 2 C u , v 2 C_(u,v)^(2)\mathbb{C}_{u, v}^{2}Cu,v2 and P x , y 1 × C u , v 2 P x , y 1 × C u , v 2 P_(x,y)^(1)xxC_(u,v)^(2)\mathbb{P}_{x, y}^{1} \times \mathbb{C}_{u, v}^{2}Px,y1×Cu,v2 via
( x : y ; u , v ) ( x : ζ a y ; ζ u , ζ 1 v ) ( x : y ; u , v ) ⟼ x : ζ a y ; ζ u , ζ − 1 v (x:y;u,v)longmapsto(x:zeta^(a)y;zeta u,zeta^(-1)v)(x: y ; u, v) \longmapsto\left(x: \zeta^{a} y ; \zeta u, \zeta^{-1} v\right)(x:y;u,v)⟼(x:ζay;ζu,ζ−1v)
where ζ = ζ n = exp ( 2 π i / n ) ζ = ζ n = exp ⁡ ( 2 Ï€ i / n ) zeta=zeta_(n)=exp(2pi i//n)\zeta=\zeta_{n}=\exp (2 \pi i / n)ζ=ζn=exp⁡(2Ï€i/n) and gcd ( n , a ) = 1 gcd ⁡ ( n , a ) = 1 gcd(n,a)=1\operatorname{gcd}(n, a)=1gcd⁡(n,a)=1. Then the projection
f : X = ( P 1 × C 2 ) / μ n Z = C 2 / μ n f : X = P 1 × C 2 / μ n ⟶ Z = C 2 / μ n f:X=(P^(1)xxC^(2))//mu_(n)longrightarrow Z=C^(2)//mu_(n)f: X=\left(\mathbb{P}^{1} \times \mathbb{C}^{2}\right) / \mu_{n} \longrightarrow Z=\mathbb{C}^{2} / \mu_{n}f:X=(P1×C2)/μn⟶Z=C2/μn
is a Q Q Q\mathbb{Q}Q-conic bundle. The variety X X XXX has exactly two singular points which are terminal cyclic quotients of type 1 n ( 1 , 1 , ± a ) 1 n ( 1 , − 1 , ± a ) (1)/(n)(1,-1,+-a)\frac{1}{n}(1,-1, \pm a)1n(1,−1,±a). The surface Z Z ZZZ has at 0 a D u D u Du\mathrm{Du}Du Val point of type A n 1 A n − 1 A_(n-1)\mathrm{A}_{\mathrm{n}-1}An−1.
McKernan proposed a natural higher-dimensional analogue of Corollary 7.2:
Conjecture 7.4. Let f : X Z f : X → Z f:X rarr Zf: X \rightarrow Zf:X→Z be a K K KKK-negative contraction such that ρ ( X / Z ) = 1 ρ ( X / Z ) = 1 rho(X//Z)=1\rho(X / Z)=1ρ(X/Z)=1 and X X XXX is ε ε epsi\varepsilonε-lc, that is, all the coefficients in (1.1.1) satisfy a i 1 + ε a i ≥ − 1 + ε a_(i) >= -1+epsia_{i} \geq-1+\varepsilonai≥−1+ε. Then Z Z ZZZ is δ δ delta\deltaδ-lc, where δ δ delta\deltaδ depends on ε ε epsi\varepsilonε and the dimension.
A deeper version of this conjecture which generalizes Theorem 5.1 and uses the notion was proposed by Shokurov. He also suggested that the optimal value of δ δ delta\deltaδ, in the
case where singularities of X X XXX are canonical and f f fff has one-dimensional fibers, equals 1 / 2 1 / 2 1//21 / 21/2. Recently, this was proved by J. Han, C. Jiang, and Y. Luo [17].
Once we have a Du Val general elephants, all extremal curve germs can be divided into two large classes which will be discussed separately in the next two sections.
Definition 7.5. Let ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) be an extremal curve germ and let f : X ( Z o ) f : X → ( Z ∋ o ) f:X rarr(Z∋o)f: X \rightarrow(Z \ni o)f:X→(Z∋o) be the corresponding contraction. Assume that the general member D | K X | D ∈ − K X D in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| is Du Val. Consider the Stein factorization:
f D : D D f ( D ) (put D = f ( D ) if f is birational). f D : D ⟶ D ′ ⟶ f ( D )  (put  D ′ = f ( D )  if  f  is birational).  f_(D):D longrightarrowD^(')longrightarrow f(D)quad" (put "D^(')=f(D)" if "f" is birational). "f_{D}: D \longrightarrow D^{\prime} \longrightarrow f(D) \quad \text { (put } D^{\prime}=f(D) \text { if } f \text { is birational). }fD:D⟶D′⟶f(D) (put D′=f(D) if f is birational). 
Then the germ ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) is said to be semistable if D D ′ D^(')D^{\prime}D′ has only (Du Val) singularities of type A n A n A_(n)\mathrm{A}_{\mathrm{n}}An. Otherwise, ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) is called exceptional.

8. SEMISTABLE GERMS

Let ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) be an irreducible extremal curve germ. By Theorem 7.1, the general member D | K X | D ∈ − K X D in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| is Du Val. In this section we assume that ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) is semistable. Excluding simple cases, we assume also that X X XXX is not Gorenstein [12] and ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) is not a Q Q Q\mathbb{Q}Q-conic bundle germ over a singular base [ 47 , 48 ] [ 47 , 48 ] [47,48][47,48][47,48]. According to Theorem 6.4, the threefold X X XXX has at most two non-Gorenstein points. Thus the following case division is natural:
Case (k1A): the set of non-Gorenstein points consists of a single point P P PPP;
Case (k2A): the set of non-Gorenstein points consists of exactly two points P 1 , P 2 P 1 , P 2 P_(1),P_(2)P_{1}, P_{2}P1,P2.
Proposition 8.1. In the above hypothesis, for the general member H | O X | H ∈ O X H in|O_(X)|H \in\left|\mathscr{O}_{X}\right|H∈|OX|, the pair ( X , H + D ) ( X , H + D ) (X,H+D)(X, H+D)(X,H+D) is lc. If, moreover, D C D ⊃ C D sup CD \supset CD⊃C, then H H HHH is normal and has only cyclic quotient singularities. In this case the singularities of H H HHH are of type T T T\mathrm{T}T.
The proof uses the inversion of adjunction [70] to extend a general hyperplane section from D D DDD to X X XXX (see [51, PROPOSITION 2.6]).
For an extremal curve germ of type (k2A), any member D | K X | D ∈ − K X D in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| contains C C CCC [38]. Hence the general hyperplane section H | O X | H ∈ O X H in|O_(X)|H \in\left|\mathscr{O}_{X}\right|H∈|OX| has only T-singularities and X X XXX can be restored as a one-parameter deformation space of H H HHH. In this case X X XXX has no singularities other than P 1 , P 2 P 1 , P 2 P_(1),P_(2)P_{1}, P_{2}P1,P2. Moreover, ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) cannot be a Q Q Q\mathbb{Q}Q-conic bundle germ [47,50]. The birational germs of type (k2A) were completely described by Mori [46]. He gave an explicit algorithm for computing divisorial contractions and flips in this case.
The structure of extremal curve germs of type (k1A) is more complicated. They were studied in [51]. In particular, the general hyperplane section H | O X | H ∈ O X H in|O_(X)|H \in\left|\mathscr{O}_{X}\right|H∈|OX| was computed. However, [51] does not provide a good description of the infinitesimal structure of X X XXX along C C CCC or an algorithm similar to [46]. This was done only in a special situation in [14]. Note that in the case ( k 1 A ) ( k 1 A ) (k1A)(\mathrm{k} 1 \mathrm{~A})(k1 A) a general member H | O X | H ∈ O X H in|O_(X)|H \in\left|\mathscr{O}_{X}\right|H∈|OX| can be nonnormal.
Examples. Similar to the example in Section 6.1, consider a surface germ H C P 1 H ⊃ C ≃ P 1 H sup C≃P^(1)H \supset C \simeq \mathbb{P}^{1}H⊃C≃P1 whose dual graph has the following graph of the minimal resolution:

7 − 7 -7-7−7 qquad\qquad 2 − 2 -2-2−2 qquad\qquad 2 − 2 -2-2−2 qquad\qquad 2 − 2 -2-2−2
where ∙ ∙\bullet∙ is a ( -1 -curve. The chain formed by white circle vertices o corresponds to a Tsingularity of type 1 25 ( 1 , 4 ) 1 25 ( 1 , 4 ) (1)/(25)(1,4)\frac{1}{25}(1,4)125(1,4). The whole configuration can be contracted to a cyclic quotient singularity H Z o H Z ∋ o H_(Z)∋oH_{Z} \ni oHZ∋o of type 1 21 ( 1 , 16 ) 1 21 ( 1 , 16 ) (1)/(21)(1,16)\frac{1}{21}(1,16)121(1,16). Since this is not a T-singularity, the induced threefold contraction must be flipping.

9. EXCEPTIONAL CURVE GERMS

In this section we assume that ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) is an exceptional irreducible extremal curve germ. As in the previous section we also assume that X X XXX is not Gorenstein and ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) is not a Q Q Q\mathbb{Q}Q-conic bundle germ over a singular base. According to the classification [ 38 , 44 , 50 ] [ 38 , 44 , 50 ] [38,44,50][38,44,50][38,44,50], the germ ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) belongs to one of following types:
  • X X XXX has a unique non-Gorenstein point P P PPP which is of type cD/2, cAx/2, cE/2, or c D / 3 c D / 3 cD//3\mathrm{cD} / 3cD/3 and ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) is of type (IA) at P P PPP;
  • X X XXX has a unique non-Gorenstein point P P PPP which is of exceptional type c A x / 4 c A x / 4 cAx//4\mathrm{cAx} / 4cAx/4 and ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) is of type (IIA), (II ) ∨ {:^(vv))\left.{ }^{\vee}\right)∨), or (IIB) at P P PPP;
  • X X XXX has a unique singular point P P PPP which is a cyclic quotient singularity of index m 5 m ≥ 5 m >= 5m \geq 5m≥5 (odd) and ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) is of type (IC) at P P PPP;
  • X X XXX has two singular points of indices m 3 m ≥ 3 m >= 3m \geq 3m≥3 (odd) and 2, then ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) is said to be of type (kAD);
  • X X XXX has three singular points of indices m 3 m ≥ 3 m >= 3m \geq 3m≥3 (odd), 2 and 1 , then ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) is said to be of type ( k 3 A ) ( k 3 A ) (k3A)(\mathrm{k} 3 \mathrm{~A})(k3 A).
In each case the general elephant is completely described in terms of its minimal resolution:
Theorem 9.1 ([38,50]). In the above hypothesis assume that the general element D | K X | D ∈ − K X D in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| contains C C CCC. Then the dual graph of ( D C ) ( D ⊃ C ) (D sup C)(D \supset C)(D⊃C) is one of the following, where white vertices ∘ @\circ∘ denote (-2)-curves on the minimal resolution of D D DDD and the black vertex ∙ ∙\bullet∙ corresponds to the proper transform of C C CCC :
(IC)
( k A D ) ( k A D ) (kAD)(\mathrm{kAD})(kAD)
(k3A)
0 0 0 1 0 ] 0 − ⋯ − 0 − 0 1 0 {:0-cdots-0-[0],[1],[0]]\left.0-\cdots-0-\begin{array}{c} 0 \\ 1 \\ 0 \end{array}\right]0−⋯−0−010]
Exceptional irreducible extremal curve germs are well studied (see [ 38 , 55 ] [ 38 , 55 ] [38,55][38,55][38,55], and references therein). For flipping ones, the general hyperplane section H | O X | H ∈ O X H in|O_(X)|H \in\left|\mathscr{O}_{X}\right|H∈|OX| is normal and has only rational singularities. It is computed in [38] and the flip is reconstructed as a oneparameter deformation space of H H HHH. For divisorial and Q Q Q\mathbb{Q}Q-conic bundle germs, the situation is more complicated. Then the general hyperplane section H H HHH can be nonnormal (see, e.g., [54]). Nevertheless, in almost all cases, except for types (kAD) and (k3A), there is a description of H | O X | H ∈ O X H in|O_(X)|H \in\left|\mathscr{O}_{X}\right|H∈|OX| and infinitesimal structure of these germs. For convenience of reference, in the table below we collect the known information on the exceptional irreducible extremal curve germs.
Type ( X , C ) ( X , C ) (X,C)(X, C)(X,C) References
index 2 germs divisorial, Q Q Q\mathbb{Q}Q-conic bundle [ 38 , § 4 ] , [ 47 , § 12 ] , [ 51 , § 7 ] [ 38 , § 4 ] , [ 47 , § 12 ] , [ 51 , § 7 ] [38,§4],[47,§12],[51,§7][38, \S 4],[47, \S 12],[51, \S 7][38,§4],[47,§12],[51,§7]
cD/3 flip, divisorial [ 38 , § 6 ] , [ 51 , § 4 ] [ 38 , § 6 ] , [ 51 , § 4 ] [38,§6],[51,§4][38, \S 6],[51, \S 4][38,§6],[51,§4]
(IC) flip, Q Q Q\mathbb{Q}Q-conic bundle (only for m = 5 ) m = 5 ) m=5)m=5)m=5) [ 38 , 88 ] , [ 52 ] [ 38 , 88 ] , [ 52 ] [38,88],[52][38,88],[52][38,88],[52]
(IIA) flip, divisorial, Q Q Q\mathbb{Q}Q-conic bundle [ 38 , § 7 ] , [ 53 , 54 ] [ 38 , § 7 ] , [ 53 , 54 ] [38,§7],[53,54][38, \S 7],[53,54][38,§7],[53,54]
(IIB) divisorial, Q Q Q\mathbb{Q}Q-conic bundle [ 52 ] [ 52 ] [52][52][52]
(II ) ∨ {:^(vv))\left.{ }^{\vee}\right)∨) divisorial, Q Q Q\mathbb{Q}Q-conic bundle [ 38 , 4.11 .2 ] , [ 47 ] [ 38 , 4.11 .2 ] , [ 47 ] [38,4.11.2],[47][38,4.11 .2],[47][38,4.11.2],[47]
(kAD) flip, divisorial, Q Q Q\mathbb{Q}Q-conic bundle [ 38 , § 9 ] , [ 45 , 47 , 50 ] [ 38 , § 9 ] , [ 45 , 47 , 50 ] [38,§9],[45,47,50][38, \S 9],[45,47,50][38,§9],[45,47,50]
(k3A) divisorial, Q Q Q\mathbb{Q}Q-conic bundle [ 38 , § 5 ] , [ 47 , 50 ] [ 38 , § 5 ] , [ 47 , 50 ] [38,§5],[47,50][38, \S 5],[47,50][38,§5],[47,50]
Type (X,C) References index 2 germs divisorial, Q-conic bundle [38,§4],[47,§12],[51,§7] cD/3 flip, divisorial [38,§6],[51,§4] (IC) flip, Q-conic bundle (only for m=5) [38,88],[52] (IIA) flip, divisorial, Q-conic bundle [38,§7],[53,54] (IIB) divisorial, Q-conic bundle [52] (II {:^(vv)) divisorial, Q-conic bundle [38,4.11.2],[47] (kAD) flip, divisorial, Q-conic bundle [38,§9],[45,47,50] (k3A) divisorial, Q-conic bundle [38,§5],[47,50]| Type | $(X, C)$ | References | | :--- | :--- | :--- | | index 2 germs | divisorial, $\mathbb{Q}$-conic bundle | $[38, \S 4],[47, \S 12],[51, \S 7]$ | | cD/3 | flip, divisorial | $[38, \S 6],[51, \S 4]$ | | (IC) | flip, $\mathbb{Q}$-conic bundle (only for $m=5)$ | $[38,88],[52]$ | | (IIA) | flip, divisorial, $\mathbb{Q}$-conic bundle | $[38, \S 7],[53,54]$ | | (IIB) | divisorial, $\mathbb{Q}$-conic bundle | $[52]$ | | (II $\left.{ }^{\vee}\right)$ | divisorial, $\mathbb{Q}$-conic bundle | $[38,4.11 .2],[47]$ | | (kAD) | flip, divisorial, $\mathbb{Q}$-conic bundle | $[38, \S 9],[45,47,50]$ | | (k3A) | divisorial, $\mathbb{Q}$-conic bundle | $[38, \S 5],[47,50]$ |
Detailed analysis of the local structure of exceptional extremal curve germs allows extending the result of Theorem 7.1 to the case of reducible central fiber containing an exceptional component:
Theorem 9.2 (Mori-Prokhorov [56]). Let ( X C ) ( X ⊃ C ) (X sup C)(X \supset C)(X⊃C) be an extremal curve germ such that C C CCC is reducible and satisfies the following condition:
( ) ( ∗ ) ^((**)){ }^{(*)}(∗) each component C i C C i ⊂ C C_(i)sub CC_{i} \subset CCi⊂C contains at most one point of index > 2 > 2 > 2>2>2.
Then the general member D | K X | D ∈ − K X D in|-K_(X)|D \in\left|-K_{X}\right|D∈|−KX| has only Du Val singularities. Moreover, for each irreducible component C i C C i ⊂ C C_(i)sub CC_{i} \subset CCi⊂C with two non-Gorenstein points or of types (IC) or (IIB), the dual graph of ( D , C i ) D , C i (D,C_(i))\left(D, C_{i}\right)(D,Ci) has the same form as the irreducible extremal curve germ ( X C i ) X ⊃ C i (X supC_(i))\left(X \supset C_{i}\right)(X⊃Ci).
The proof uses the extension techniques of sections of | K X | − K X |-K_(X)|\left|-K_{X}\right||−KX| from a good member S | 2 K X | S ∈ − 2 K X S in|-2K_(X)|S \in\left|-2 K_{X}\right|S∈|−2KX| (see Section 7.2).

10. Q Q Q\mathbb{Q}Q-FANO THREEFOLDS

In arbitrary dimension, Q Q Q\mathbb{Q}Q-Fano threefolds are bounded, i.e., they are contained in fibers of a morphism of schemes of finite type. This is a consequence of the much more general fact [4]. In dimension 3, there are effective results based on the orbifold RiemannRoch formula (4.1.1) and Bogomolov-Miyaoka inequality applied to the restriction of the
reflexive sheaf ( Ω X 1 ) Ω X 1 ∨ ∨ (Omega_(X)^(1))^(vv vv)\left(\Omega_{X}^{1}\right)^{\vee \vee}(ΩX1)∨∨ to a sufficiently general hyperplane section [33]. In particular, combining (4.1.1) with Serre duality, we obtain
χ ( O X ) = 1 24 ( K X c 2 ( X ) + P ( m P 1 m p ) ) χ O X = 1 24 − K X â‹… c 2 ( X ) + ∑ P   m P − 1 m p chi(O_(X))=(1)/(24)(-K_(X)*c_(2)(X)+sum_(P)(m_(P)-(1)/(m_(p))))\chi\left(\mathscr{O}_{X}\right)=\frac{1}{24}\left(-K_{X} \cdot \mathrm{c}_{2}(X)+\sum_{P}\left(m_{P}-\frac{1}{m_{p}}\right)\right)χ(OX)=124(−KXâ‹…c2(X)+∑P(mP−1mp))
where m P m P m_(P)m_{P}mP is the index of a virtual quotient singularity of X X XXX [66]. Since X X XXX is Q Q Q\mathbb{Q}Q-Fano, by Kawamata-Viehweg vanishing theorem [35], one has χ ( O X ) = 1 χ O X = 1 chi(O_(X))=1\chi\left(\mathscr{O}_{X}\right)=1χ(OX)=1. Arguments based on Bogomolov-Miyaoka inequality show that K X c 2 ( X ) − K X â‹… c 2 ( X ) -K_(X)*c_(2)(X)-K_{X} \cdot \mathrm{c}_{2}(X)−KXâ‹…c2(X) is positive (see [33]). This gives an effective bound of the indices of singularities of X X XXX. Similarly, one can get an upper bound of the anticanonical degree K X 3 − K X 3 -K_(X)^(3)-K_{X}^{3}−KX3. Moreover, analyzing the methods of [33], it is possible to enumerate Hilbert series of all Q Q Q\mathbb{Q}Q-Fano threefolds. This information is collected in [6] in a form of a huge computer database of possible "candidates." It was extensively explored by many authors, basically to obtain lists of examples representing Q Q Q\mathbb{Q}Q-Fano threefolds as subvarieties of small codimension in a weighted projective space (see, e.g., [ 7 , 21 ] [ 7 , 21 ] [7,21][7,21][7,21] and references therein).
Examples. - There are 130 (resp. 125) families of Q Q Q\mathbb{Q}Q-Fano threefolds that are representable as hypersurfaces (resp. codimension 2 complete intersections) in weighted projective spaces [ 6 , 21 ] [ 6 , 21 ] [6,21][6,21][6,21].
  • Toric Q Q Q\mathbb{Q}Q-Fano threefolds are exactly weighted projective spaces P ( 3 , 4 , 5 , 7 ) P ( 3 , 4 , 5 , 7 ) P(3,4,5,7)\mathbb{P}(3,4,5,7)P(3,4,5,7), P ( 2 , 3 , 5 , 7 ) , P ( 1 , 3 , 4 , 5 ) , P ( 1 , 2 , 3 , 5 ) , P ( 1 , 1 , 2 , 3 ) , P ( 1 , 1 , 1 , 2 ) , P 3 = P ( 1 , 1 P ( 2 , 3 , 5 , 7 ) , P ( 1 , 3 , 4 , 5 ) , P ( 1 , 2 , 3 , 5 ) , P ( 1 , 1 , 2 , 3 ) , P ( 1 , 1 , 1 , 2 ) , P 3 = P ( 1 , 1 P(2,3,5,7),P(1,3,4,5),P(1,2,3,5),P(1,1,2,3),P(1,1,1,2),P^(3)=P(1,1\mathbb{P}(2,3,5,7), \mathbb{P}(1,3,4,5), \mathbb{P}(1,2,3,5), \mathbb{P}(1,1,2,3), \mathbb{P}(1,1,1,2), \mathbb{P}^{3}=\mathbb{P}(1,1P(2,3,5,7),P(1,3,4,5),P(1,2,3,5),P(1,1,2,3),P(1,1,1,2),P3=P(1,1, 1 , 1 ) 1 , 1 ) 1,1)1,1)1,1), and the quotient of P 3 P 3 P^(3)\mathbb{P}^{3}P3 by μ 5 μ 5 mu_(5)\boldsymbol{\mu}_{5}μ5 that acts diagonally with weights ( 1 , 2 , 3 , 4 ) [ 6 ] ( 1 , 2 , 3 , 4 ) [ 6 ] (1,2,3,4)[6](1,2,3,4)[6](1,2,3,4)[6].
Although the classification is very far from completion, there are several systematic results. For example, the optimal upper bound of the degree K X 3 − K X 3 -K_(X)^(3)-K_{X}^{3}−KX3 of Q Q Q\mathbb{Q}Q-Fano threefolds was obtained in [58]. If X X XXX is singular, it is equal to 125/2 and achieved for the weighted projective space P ( 1 , 1 , 1 , 2 ) P ( 1 , 1 , 1 , 2 ) P(1,1,1,2)\mathbb{P}(1,1,1,2)P(1,1,1,2). The lower bound of the degree equals 1 / 330 1 / 330 1//3301 / 3301/330 [8] and is achieved for a hypersurface of degree 66 in P ( 1 , 5 , 6 , 22 , 33 ) P ( 1 , 5 , 6 , 22 , 33 ) P(1,5,6,22,33)\mathbb{P}(1,5,6,22,33)P(1,5,6,22,33). It is known that, under certain conditions, General Elephant Conjecture 3.1 holds for Q Q Q\mathbb{Q}Q-Fano threefolds modulo deformations [67].

ACKNOWLEDGMENTS

The author would like to thank Professors Shigefumi Mori and Vyacheslav Shokurov for helpful comments on the original version of this paper.

FUNDING

This work was performed at the Steklov International Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2019-1614).

REFERENCES

[1] V. Alexeev, General elephants of Q-Fano 3-folds. Compos. Math. 91 (1994), no. 1, 91 116 91 − 116 91-11691-11691−116.
[2] V. Alexeev and V. V. Nikulin, Del Pezzo and K3 surfaces. MSJ Mem. 15, Mathematical Society of Japan, Tokyo, 2006.
[3] C. Birkar, Birational geometry of algebraic varieties. In Proceedings of the International Congress of Mathematicians-Rio de Janeiro 2018. Vol. II. Invited lectures, pp. 565-590, World Sci. Publ., Hackensack, NJ, 2018.
[4] C. Birkar, Singularities of linear systems and boundedness of Fano varieties. Ann. of Math. (2) 193 (2021), no. 2, 347-405.
[5] C. Birkar, P. Cascini, C. D. Hacon, and J. McKernan, Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. 23 (2010), no. 2, 405-468.
[6] G. Brown, et al., Graded ring database, http://www.grdb.co.uk/.
[7] G. Brown, M. Kerber, and M. Reid, Fano 3-folds in codimension 4, Tom and Jerry. Part I. Compos. Math. 148 (2012), no. 4, 1171-1194.
[8] J. A. Chen and M. Chen, An optimal boundedness on weak Q Q Q\mathbf{Q}Q-Fano 3-folds. Adv. Math. 219 (2008), no. 6, 2086-2104.
[9] A. Corti, Factoring birational maps of threefolds after Sarkisov. J. Algebraic Geom. 4 (1995), no. 2, 223-254.
[10] A. Corti, Singularities of linear systems and 3-fold birational geometry. In Explicit birational geometry of 3-folds, pp. 259-312, London Math. Soc. Lecture Note Ser. 281, Cambridge Univ. Press, Cambridge, 2000.
[11] A. Corti, A. Pukhlikov, and M. Reid, Fano 3-fold hypersurfaces. In Explicit birational geometry of 3-folds, pp. 175-258, London Math. Soc. Lecture Note Ser. 281, Cambridge Univ. Press, Cambridge, 2000.
[12] S. Cutkosky, Elementary contractions of Gorenstein threefolds. Math. Ann. 280 (1988), no. 3, 521-525.
[13] P. Hacking and Y. Prokhorov, Smoothable del Pezzo surfaces with quotient singularities. Compos. Math. 146 (2010), no. 1, 169-192.
[14] P. Hacking, J. Tevelev, and G. Urzúa, Flipping surfaces. J. Algebraic Geom. 26 (2017), no. 2, 279-345.
[15] C. D. Hacon and J. McKernan, Existence of minimal models for varieties of log general type. II. J. Amer. Math. Soc. 23 (2010), no. 2, 469-490.
[16] C. D. Hacon and J. McKernan, The Sarkisov program. J. Algebraic Geom. 22 (2013), no. 2, 389-405.
[17] J. Han, C. Jiang, and Y. Luo, Shokurov's conjecture on conic bundles with canonical singularities. 2021, arXiv:2104.15072.
[18] T. Hayakawa, Blowing ups of 3-dimensional terminal singularities. Publ. Res. Inst. Math. Sci. 35 (1999), no. 3, 515-570.
[19] T. Hayakawa, Blowing ups of 3-dimensional terminal singularities. II. Publ. Res. Inst. Math. Sci. 36 (2000), no. 3, 423-456.
[20] T. Hayakawa, Divisorial contractions to 3-dimensional terminal singularities with discrepancy one. J. Math. Soc. Japan 57 (2005), no. 3, 651-668.
[21] A. R. Iano-Fletcher, Working with weighted complete intersections. In Explicit birational geometry of 3-folds, pp. 101-173, London Math. Soc. Lecture Note Ser. 281, Cambridge Univ. Press, Cambridge, 2000.
[22] V. A. Iskovskih, Fano threefolds. I. Izv. Ross. Akad. Nauk Ser. Mat. 41 (1977), no. 3, 516-562, 717 .
[23] V. A. Iskovskih, Fano threefolds. II. Izv. Ross. Akad. Nauk Ser. Mat. 42 (1978), no. 3, 506-549.
[24] V. A. Iskovskikh, On a rationality criterion for conic bundles. Sb. Math. 187 (1996), no. 7, 1021-1038.
[25] V. A. Iskovskikh and Y. Prokhorov, Fano varieties. Algebraic geometry V. Encyclopaedia Math. Sci. 47, Springer, Berlin, 1999.
[26] M. Kawakita, Divisorial contractions in dimension three which contract divisors to smooth points. Invent. Math. 145 (2001), no. 1, 105-119.
[27] M. Kawakita, Divisorial contractions in dimension three which contract divisors to compound A 1 A 1 A_(1)A_{1}A1 points. Compos. Math. 133 (2002), no. 1, 95-116.
[28] M. Kawakita, General elephants of three-fold divisorial contractions. J. Amer. Math. Soc. 16 (2003), no. 2, 331-362.
[29] M. Kawakita, Three-fold divisorial contractions to singularities of higher indices. Duke Math. J. 130 (2005), no. 1, 57-126.
[30] M. Kawakita, Supplement to classification of threefold divisorial contractions. Nagoya Math. J. 206 (2012), 67-73.
[31] M. Kawakita, The index of a threefold canonical singularity. Amer. J. Math. 137 (2015), no. 1, 271-280.
[32] Y. Kawamata, Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces. Ann. of Math. (2) 127 (1988), no. 1, 93-163.
[33] Y. Kawamata, Boundedness of Q-Fano threefolds. In Proceedings of the International Conference on Algebra, Part 3 (Novosibirsk, 1989), pp. 439-445, Contemp. Math. 131, Amer. Math. Soc., Providence, RI, 1992.
[34] Y. Kawamata, Divisorial contractions to 3-dimensional terminal quotient singularities. In Higher-dimensional complex varieties (Trento, 1994), pp. 241-246, de Gruyter, Berlin, 1996.
[35] Y. Kawamata, K. Matsuda, and K. Matsuki, Introduction to the minimal model problem. In Algebraic geometry (Sendai, 1985), pp. 283-360, Adv. Stud. Pure Math. 10, North-Holland, Amsterdam, 1987.
[36] K. Kodaira, On compact analytic surfaces. II, III. Ann. of Math. (2) 77 (1963), 563-626; ibid. 78 (1963), 1-40
[37] J. Kollár, Real algebraic threefolds. III. Conic bundles. J. Math. Sci. (N. Y.) 94 (1999), no. 1, 996-1020.
[38] J. Kollár and S. Mori, Classification of three-dimensional flips. J. Amer. Math. Soc. 5 (1992), no. 3, 533-703.
[39] J. Kollár and S. Mori, Birational geometry of algebraic varieties. Cambridge Tracts in Math. 134, Cambridge University Press, Cambridge, 1998.
[40] J. Kollár and N. I. Shepherd-Barron, Threefolds and deformations of surface singularities. Invent. Math. 91 (1988), no. 2, 299-338.
[41] M. Manetti, Normal degenerations of the complex projective plane. J. Reine Angew. Math. 419 (1991), 89-118.
[42] S. Mori, Threefolds whose canonical bundles are not numerically effective. Ann. of Math. (2) 116 (1982), 133-176.
[43] S. Mori, On 3-dimensional terminal singularities. Nagoya Math. J. 98 (1985), 43-66.
[44] S. Mori, Flip theorem and the existence of minimal models for 3-folds. J. Amer. Math. Soc. 1 (1988), no. 1, 117-253.
[45] S. Mori, Errata to: "Classification of three-dimensional flips". [J. Amer. Math. Soc. 5 (1992), no. 3, 533-703; mr1149195] by J. Kollár and S. Mori, J. Amer. Math. Soc. 20 (2007), no. 1, 269-271.
[46] S. Mori, On semistable extremal neighborhoods. In Higher dimensional birational geometry (Kyoto, 1997), pp. 157-184, Adv. Stud. Pure Math. 35, Math. Soc. Japan, Tokyo, 2002.
[47] S. Mori and Y. Prokhorov, On Q-conic bundles. Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 315-369.
[48] S. Mori and Y. Prokhorov, On Q-conic bundles. II. Publ. Res. Inst. Math. Sci. 44 (2008), no. 3, 955-971.
[49] S. Mori and Y. Prokhorov, Multiple fibers of del Pezzo fibrations. Proc. Steklov Inst. Math. 264 (2009), no. 1, 131-145.
[50] S. Mori and Y. Prokhorov, On Q-conic bundles, III. Publ. Res. Inst. Math. Sci. 45 (2009), no. 3, 787-810.
[51] S. Mori and Y. Prokhorov, Threefold extremal contractions of type IA. Kyoto J. Math. 51 (2011), no. 2, 393-438.
[52] S. Mori and Y. Prokhorov, Threefold extremal contractions of types (IC) and (IIB). Proc. Edinb. Math. Soc. 57 (2014), no. 1, 231-252.
[53] S. Mori and Y. Prokhorov, Threefold extremal contractions of type (IIA), I. Izv. Math. 80 (2016), no. 5, 884-909.
[54] S. Mori and Y. Prokhorov, Threefold extremal contractions of type (IIA), II. In Geometry and physics: a festschrift in honour of Nigel Hitchin: volume 2, edited by A. Dancer, J. E. Andersen, and O. Garcia-Prada, Oxford University Press, 2018 .
[55] S. Mori and Y. Prokhorov, Threefold extremal curve germs with one nonGorenstein point. Izv. Math. 83 (2019), no. 3, 565-612.
[56] S. Mori and Y. G. Prokhorov, General elephants for threefold extremal contractions with one-dimensional fibres: exceptional case. Mat. Sb. 212 (2021), no. 3, 88 111 88 − 111 88-11188-11188−111.
[57] Y. Namikawa, Smoothing Fano 3-folds. J. Algebraic Geom. 6 (1997), no. 2, 307-324.
[58] Y. Prokhorov, The degree of Q-Fano threefolds. Sb. Math. 198 (2007), no. 11, 153 174 153 − 174 153-174153-174153−174.
[59] Y. Prokhorov, Q-Fano threefolds of large Fano index, I. Doc. Math. 15 (2010), 843-872.
[60] Y. Prokhorov, A note on degenerations of del Pezzo surfaces. Ann. Inst. Fourier 65 (2015), no. 1, 369-388.
[61] Y. Prokhorov, The rationality problem for conic bundles. Russian Math. Surveys 73 (2018), no. 3, 375-456.
[62] Y. Prokhorov, Log canonical degenerations of del Pezzo surfaces in Q-Gorenstein families. Kyoto J. Math. 59 (2019), no. 4, 1041-1073.
[63] Y. Prokhorov, Equivariant minimal model program. Russian Math. Surveys 76 (2021), no. 3, 461-542.
[64] Y. G. Prokhorov, Lectures on complements on log surfaces, MSJ Mem. 10, Mathematical Society of Japan, Tokyo, 2001.
[65] M. Reid, Minimal models of canonical 3-folds. In Algebraic varieties and analytic varieties (Tokyo, 1981), pp. 131-180, Adv. Stud. Pure Math. 1, 1981, NorthHolland, Amsterdam, 1983.
[66] M. Reid, Young person's guide to canonical singularities. In Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), pp. 345-414, Proc. Sympos. Pure Math. 46, Amer. Math. Soc., Providence, RI, 1987.
[67] T. Sano, Deforming elephants of Q Q Q\mathbb{Q}Q-Fano 3-folds. J. Lond. Math. Soc. (2) 95 (2017), no. 1, 23-51.
[68] V. Shokurov and S. R. Choi, Geography of log models: theory and applications. Cent. Eur. J. Math. 9 (2011), no. 3, 489-534.
[69] V. V. Shokurov, A nonvanishing theorem. Math. USSR, Izv. 26 (1986), no. 3, 591-604.
[70] V. V. Shokurov, 3-fold log flips. Izv. Math. 40 (1993), no. 1, 95-202.
[71] N. Tziolas, Terminal 3-fold divisorial contractions of a surface to a curve. I. Compos. Math. 139 (2003), no. 3, 239-261.
[72] N. Tziolas, Families of D D DDD-minimal models and applications to 3-fold divisorial contractions. Proc. Lond. Math. Soc. (3) 90 (2005), no. 2, 345-370.
[73] N. Tziolas, Three dimensional divisorial extremal neighborhoods. Math. Ann. 333 (2005), no. 2, 315-354.
[74] N. Tziolas, Three-fold divisorial extremal neighborhoods over c E 7 c E 7 cE_(7)c E_{7}cE7 and c E 6 c E 6 cE_(6)c E_{6}cE6 compound DuVal singularities. Internat. J. Math. 21 (2010), no. 1, 1-23.

YURI PROKHOROV

Steklov Mathematical Institute, 8 Gubkina street, Moscow 119991, Russia, prokhoro @mi-ras.ru

SOME ASPECTS OF RATIONAL POINTS AND RATIONAL CURVES

OLIVIER WITTENBERG

ABSTRACT

Various methods have been used to construct rational points and rational curves on rationally connected algebraic varieties. We survey recent advances in two of them, the descent and the fibration method, in a number-theoretical context (rational points over number fields) and in an algebro-geometric one (rational curves on real varieties), and discuss the question of rational points over function fields of p p ppp-adic curves.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 11G35; Secondary 14G05, 14C25, 14P99

KEYWORDS

Rational points, rational curves, inverse Galois problem, tight approximation, p p ppp-adic function fields

1. INTRODUCTION

Let X X XXX be an algebraic variety over a field k k kkk and X ( k ) X ( k ) X(k)X(k)X(k) the set of its rational points.
The search for explicit descriptions of the set X ( k ) X ( k ) X(k)X(k)X(k) when k k kkk is a number field is one of the oldest themes of number theory. A modern point of view on this problem consists in embedding X ( k ) X ( k ) X(k)X(k)X(k) diagonally into the topological space X ( A k ) X A k X(A_(k))X\left(\mathbf{A}_{k}\right)X(Ak) of adelic points of X X XXX and attempting to identify its topological closure. By general principles that were formulated by Lang after the works of Mordell, Weil, and Siegel, the answer is expected to depend in a crucial manner on the geometry of X X XXX. For instance, assuming that X X XXX is smooth and projective and that an embedding k C k ↪ C k↪Ck \hookrightarrow \mathbf{C}k↪C is given, the set X ( k ) X ( k ) X(k)X(k)X(k) is conjectured to be finite if the complex variety X C X C X_(C)X_{\mathbf{C}}XC is hyperbolic (see [71]). One may then seek to count, list, or bound its elements. At the other end of the spectrum, if X C X C X_(C)X_{\mathbf{C}}XC is a rationally connected smooth projective variety in the sense of Campana [12] and Kollár-Miyaoka-Mori [69], then one expects that the set X ( k ) X ( k ) X(k)X(k)X(k) is Zariski dense in X X XXX whenever it is nonempty; more precisely, by a conjecture of Colliot-Thélène, the closure of X ( k ) X ( k ) X(k)X(k)X(k) in X ( A k ) X A k X(A_(k))X\left(\mathbf{A}_{k}\right)X(Ak) should coincide in this case with the BrauerManin set X ( A k ) Br ( X ) X A k Br ⁡ ( X ) X(A_(k))^(Br(X))X\left(\mathbf{A}_{k}\right)^{\operatorname{Br}(X)}X(Ak)Br⁡(X) defined by Manin [76]. This far-reaching conjecture encompasses in particular the inverse Galois problem, and its refinement the Grunwald problem (see [ 28 , 33 [ 28 , 33 [28,33[28,33[28,33, 36 ] 36 ] 36]36]36], [ 95 , $ 3.5 ] [ 95 , $ 3.5 ] [95,$3.5][95, \$ 3.5][95,$3.5].
Criteria for the existence of rational points on X X XXX are also of relevance outside of number theory, when k k kkk is no longer assumed to be a number field. For instance, the GraberHarris-Starr theorem [34], a central result in the theory of rational curves on complex algebraic varieties, is equivalent to the statement that X ( k ) X ( k ) ≠ ∅ X(k)!=O/X(k) \neq \varnothingX(k)≠∅ if k k kkk is the function field of a complex curve and X X XXX is a rationally connected variety. (We say that X X XXX is rationally connected to mean that for any algebraically closed field extension K K KKK of k k kkk, the variety X K X K X_(K)X_{K}XK over K K KKK is rationally connected in the sense of [ 12 , 69 ] [ 12 , 69 ] [12,69][12,69][12,69].) As another example, if X X XXX is a real algebraic variety with no real point and k k kkk denotes the function field of the real conic given by x 2 + y 2 = 1 x 2 + y 2 = − 1 x^(2)+y^(2)=-1x^{2}+y^{2}=-1x2+y2=−1, the existence of a geometrically rational curve on X X XXX-a property conjectured by Kollár to hold whenever X X XXX is a positive-dimensional rationally connected variety-is equivalent to the statement that X ( k ) X ( k ) ≠ ∅ X(k)!=O/X(k) \neq \varnothingX(k)≠∅.
The results we discuss in this expository article concern the existence of rational points in two very distinct contexts, leading to the following two concrete theorems, obtained in collaboration with Yonatan Harpaz and with Olivier Benoist, respectively. As we shall see, their proofs roughly follow, perhaps somewhat surprisingly, a common general strategy.
Theorem A (see [50]). Let G G GGG be a finite nilpotent group. Let k k kkk be a number field.
(1) There exist Galois extensions K / k K / k K//kK / kK/k with Galois group G G GGG.
(2) If v 1 , , v n v 1 , … , v n v_(1),dots,v_(n)v_{1}, \ldots, v_{n}v1,…,vn are pairwise distinct places of k k kkk none of which is a finite place dividing the order of G G GGG, and w 1 , , w n w 1 , … , w n w_(1),dots,w_(n)w_{1}, \ldots, w_{n}w1,…,wn are places of K K KKK above v 1 , , v n v 1 , … , v n v_(1),dots,v_(n)v_{1}, \ldots, v_{n}v1,…,vn, then in (1), one can require that the extensions K w i / k v i K w i / k v i K_(w_(i))//k_(v_(i))K_{w_{i}} / k_{v_{i}}Kwi/kvi be isomorphic to any prescribed collection of Galois extensions of k v 1 , , k v n k v 1 , … , k v n k_(v_(1)),dots,k_(v_(n))k_{v_{1}}, \ldots, k_{v_{n}}kv1,…,kvn whose Galois groups are subgroups of G G GGG.
Theorem B (see [6]). Let X X XXX be a smooth, proper variety over R R R\mathbf{R}R. Let ε : S 1 X ( R ) ε : S 1 → X ( R ) epsi:S^(1)rarr X(R)\varepsilon: \mathbf{S}^{1} \rightarrow X(\mathbf{R})ε:S1→X(R) be a continuous map. Assume that X X XXX is birationally equivalent to a homogeneous space of a linear algebraic group over R R R\mathbf{R}R. Then there exist morphisms of algebraic varieties P R 1 X P R 1 → X P_(R)^(1)rarr X\mathbf{P}_{\mathbf{R}}^{1} \rightarrow XPR1→X that induce maps P 1 ( R ) = S 1 X ( R ) P 1 ( R ) = S 1 → X ( R ) P^(1)(R)=S^(1)rarr X(R)\mathbf{P}^{1}(\mathbf{R})=\mathbf{S}^{1} \rightarrow X(\mathbf{R})P1(R)=S1→X(R) arbitrarily close to ε ε epsi\varepsilonε in the compact-open topology.
Theorem A (1) was first proved by Shafarevich in his seminal work on the inverse Galois problem for solvable groups (see [82, CHAPTER IX, § 6]; it should be noted that nilpotent groups form the most difficult case in his proof); the proof given in [50] is independent from his and has a geometric flavour. Theorem A (2), on the other hand, was new in [50] and was not accessible with Shafarevich's methods.
As far as we know, Theorem B might hold under the sole assumption that X X XXX is rationally connected. This is a question we raise in [6]. Theorem B provides the first examples of a positive answer to it for varieties that are not R R R\mathbf{R}R-rational (indeed, not even C C C\mathbf{C}C-rational). For R R R\mathbf{R}R-rational varieties, the conclusion of Theorem B was previously shown, by Bochnak and Kucharz [8], to follow from the Stone-Weierstrass theorem.
The first step in the proofs of Theorems A and B consists in strengthening and reformulating the desired conclusion in terms of the existence of suitable rational points on suitable varieties over suitable fields. In the case of Theorem A, the varieties in question are homogeneous spaces of S L n S L n SL_(n)\mathrm{SL}_{n}SLn over number fields; for the proof, though not for the statement, it is crucial to not restrict to homogeneous spaces that have rational points (i.e., to homogeneous spaces of the form S L n / G ) S L n / G {:SL_(n)//G)\left.\mathrm{SL}_{n} / G\right)SLn/G). In the case of Theorem B, the varieties in question are homogeneous spaces of linear algebraic groups, over the rational function field R ( t ) R ( t ) R(t)\mathbf{R}(t)R(t); for the proof, though not for the statement, it is crucial to not restrict to homogeneous spaces or algebraic groups that are defined over R R R\mathbf{R}R. In the remainder of the proofs of Theorems A and B B B\mathrm{B}B, one establishes the validity of these strengthened formulations by combining geometric dévissages of the underlying algebraic varieties with two general tools: the descent method and the fibration method. The fibration method, whose first instance can be found in the work of Hasse on the local-global principle for quadratic forms, consists in reducing the desired property for a variety V V VVV endowed with a morphism p : V B p : V → B p:V rarr Bp: V \rightarrow Bp:V→B with geometrically irreducible generic fibre to the same property for B B BBB and for a collection of smooth fibres of p p ppp. The descent method, which goes back to Fermat, attempts to reduce the desired property for a variety V V VVV endowed with a torsor p : W V p : W → V p:W rarr Vp: W \rightarrow Vp:W→V under a (possibly disconnected) linear algebraic group over k k kkk to the same property for W W WWW and for all of its twists. It was developed in the context of elliptic curves, for torsors under finite abelian groups, by Mordell, Cassels, and Tate, and the setup was later extended to torsors under positive-dimensional linear algebraic groups by Colliot-Thélène and Sansuc, Skorobogatov, Harari.
We take Theorems A and B as excuses leading us to the general study of rational points on rationally connected varieties defined over number fields or over function fields of real curves. We discuss recent advances in the fibration and descent methods in these two contexts in Sections 2 and 3, stating along the way the main open questions that surround Theorems A and B and their proofs. We then turn, in Section 4, to function fields of p p ppp-adic curves, and speculate about the existence of a p p ppp-adic analogue of the "tight approximation"
property discussed in Section 3 that would enable one to exploit fibration and descent methods in the study of rational curves over p p ppp-adic fields and more generally of rational points over function fields of p p ppp-adic curves.

2. SOLVABLE GROUPS AND THE GRUNWALD PROBLEM IN INVERSE GALOIS THEORY

2.1. Homogeneous spaces

It is the following general theorem about the arithmetic of homogeneous spaces of linear algebraic groups that underlies Theorem A.
Theorem 2.1. Let V V VVV be a homogeneous space of a connected linear algebraic group L L LLL over a number field k k kkk. Let X X XXX be a smooth compactification of V V VVV. Let v ¯ V ( k ¯ ) v ¯ ∈ V ( k ¯ ) bar(v)in V( bar(k))\bar{v} \in V(\bar{k})v¯∈V(k¯). Assume that the group of connected components G G GGG of the stabiliser of v ¯ v ¯ bar(v)\bar{v}v¯ is supersolvable, in the sense that it possesses a normal series 1 = G 0 G m = G 1 = G 0 â—ƒ ⋯ â—ƒ G m = G 1=G_(0)â—ƒcdotsâ—ƒG_(m)=G1=G_{0} \triangleleft \cdots \triangleleft G_{m}=G1=G0◃⋯◃Gm=G such that the quotients G i + 1 / G i G i + 1 / G i G_(i+1)//G_(i)G_{i+1} / G_{i}Gi+1/Gi are cyclic while the subgroups G i G i G_(i)G_{i}Gi are normal in G G GGG and are stable under the natural outer action of Gal ( k ¯ / k ) Gal ⁡ ( k ¯ / k ) Gal( bar(k)//k)\operatorname{Gal}(\bar{k} / k)Gal⁡(k¯/k) on G G GGG. Then the subset X ( k ) X ( k ) X(k)X(k)X(k) is dense in X ( A k ) Br ( X ) X A k Br ⁡ ( X ) X(A_(k))^(Br(X))X\left(\mathbf{A}_{k}\right)^{\operatorname{Br}(X)}X(Ak)Br⁡(X).
Here and elsewhere, by "compactification of V V VVV," we mean a proper variety over k k kkk that contains V V VVV as a dense open subset; we do not require that the algebraic group L L LLL act on the compactification. Examples of supersolvable groups with respect to the trivial outer action of Gal ( k ¯ / k ) Gal ⁡ ( k ¯ / k ) Gal( bar(k)//k)\operatorname{Gal}(\bar{k} / k)Gal⁡(k¯/k) include finite nilpotent groups and dihedral groups. With a nontrivial outer action of Gal ( k ¯ / k ) Gal ⁡ ( k ¯ / k ) Gal( bar(k)//k)\operatorname{Gal}(\bar{k} / k)Gal⁡(k¯/k), however, even abelian groups need not be supersolvable. Previous work of Borovoi [10] nevertheless establishes the conclusion of Theorem 2.1 in many cases where the stabiliser of v ¯ v ¯ bar(v)\bar{v}v¯ is abelian but not necessarily supersolvable.
Theorem 2.1 can be found in [50, THÉORÄ–ME B] in the particular case where L L LLL is semi-simple simply connected and the stabiliser of v ¯ v ¯ bar(v)\bar{v}v¯ is finite, and in [51, coroLLARY 4.5] in general. To deduce Theorem A from it, embed G G GGG into S L n ( k ) S L n ( k ) SL_(n)(k)\mathrm{SL}_{n}(k)SLn(k) for some n n nnn, take L = S L n L = S L n L=SL_(n)L=\mathrm{SL}_{n}L=SLn and V = S L n / G V = S L n / G V=SL_(n)//GV=\mathrm{SL}_{n} / GV=SLn/G and let H H HHH denote the set of points of V V VVV above which the fibre of the étale cover π : L V Ï€ : L → V pi:L rarr V\pi: L \rightarrow VÏ€:L→V is irreducible. The function field of the fibre of π Ï€ pi\piÏ€ above any rational point contained in H H HHH is a Galois extension of k k kkk with Galois group G G GGG. On the other hand, by a theorem of Ekedahl [33], the density of X ( k ) X ( k ) X(k)X(k)X(k) in X ( A k ) Br ( X ) X A k Br ⁡ ( X ) X(A_(k))^(Br(X))X\left(\mathbf{A}_{k}\right)^{\operatorname{Br}(X)}X(Ak)Br⁡(X) implies that of X ( k ) H X ( k ) ∩ H X(k)nn HX(k) \cap HX(k)∩H in X ( A k ) Br ( X ) X A k Br ⁡ ( X ) X(A_(k))^(Br(X))X\left(\mathbf{A}_{k}\right)^{\operatorname{Br}(X)}X(Ak)Br⁡(X). Thus, Theorem 2.1 ensures the existence of Galois extensions K / k K / k K//kK / kK/k with Galois group G G GGG having a local behaviour prescribed by any element of the Brauer-Manin set
X ( A k ) Br ( X ) X A k Br ⁡ ( X ) X(A_(k))^(Br(X))X\left(\mathbf{A}_{k}\right)^{\operatorname{Br}(X)}X(Ak)Br⁡(X); that is, one may freely prescribe the completions of K K KKK at any finite set of places of k k kkk, as long as these prescriptions satisfy a certain global reciprocity condition determined by Br ( X ) Br ⁡ ( X ) Br(X)\operatorname{Br}(X)Br⁡(X). By a theorem of Lucchini Arteche [75, 86 6, this reciprocity condition imposes, in fact, no restriction at the places indicated in Theorem A (2).

2.2. Geometry

In the special case where L = S L n L = S L n L=SL_(n)L=\mathrm{SL}_{n}L=SLn and the stabiliser of v ¯ v ¯ bar(v)\bar{v}v¯ is a finite group G G GGG, the geometry behind the proof of Theorem 2.1 can be summarised with the following assertion:
there exist an algebraic torus T T TTT over k k kkk and a torsor Y ¯ X k ¯ Y ¯ → X k ¯ bar(Y)rarrX_( bar(k))\bar{Y} \rightarrow X_{\bar{k}}Y¯→Xk¯ under T k ¯ T k ¯ T_( bar(k))T_{\bar{k}}Tk¯ whose isomorphism class is invariant under Gal ( k ¯ / k ) Gal ⁡ ( k ¯ / k ) Gal( bar(k)//k)\operatorname{Gal}(\bar{k} / k)Gal⁡(k¯/k), such that for any torsor Y X Y → X Y rarr XY \rightarrow XY→X under T T TTT whose base change to X k ¯ X k ¯ X_( bar(k))X_{\bar{k}}Xk¯ is isomorphic to Y ¯ Y ¯ bar(Y)\bar{Y}Y¯, there exist a dense open subset W Y W ⊆ Y W sube YW \subseteq YW⊆Y and a smooth morphism p : W Q p : W → Q p:W rarr Qp: W \rightarrow Qp:W→Q to a quasitrivial torus Q Q QQQ (i.e., a torus of the form R E / k G m R E / k G m R_(E//k)G_(m)R_{E / k} \mathbf{G}_{\mathrm{m}}RE/kGm for a nonzero étale k k kkk-algebra E E EEE ) whose fibres are homogeneous spaces of S L n S L n SL_(n)\mathrm{SL}_{n}SLn with geometric stabiliser isomorphic to G m 1 G m − 1 G_(m-1)G_{m-1}Gm−1. In addition, the morphism p p ppp admits a rational section over k ¯ k ¯ bar(k)\bar{k}k¯.
This geometry is the key to a proof of Theorem 2.1 by an induction on m m mmm, at each step of which one applies the descent method and the fibration method, in the form of Theorems 2.2 and 2.3 below. It should be noted that even if G G GGG is embedded into S L n ( k ) S L n ( k ) SL_(n)(k)\mathrm{SL}_{n}(k)SLn(k) and V = S L n / G V = S L n / G V=SL_(n)//GV=\mathrm{SL}_{n} / GV=SLn/G, the homogeneous spaces of S L n S L n SL_(n)\mathrm{SL}_{n}SLn that arise as fibres of p p ppp need not possess rational points. Thus, for the induction to be possible, one cannot restrict the statement of Theorem 2.1 to homogeneous spaces of the form S L n / G S L n / G SL_(n)//G\mathrm{SL}_{n} / GSLn/G, even though only homogeneous spaces of this form are relevant for Theorem A.

2.3. Descent

The following theorem, which was established in [50] and can also be deduced from [13], is the definitive statement of descent theory in the case of smooth and proper rationally connected varieties over number fields. For geometrically rational X X XXX, this theorem is due to Colliot-Thélène and Sansuc [25]. The homogeneous spaces of Theorem 2.1 are not geometrically rational in general (Saltman, Bogomolov; see [26]).
Theorem 2.2. Let X X XXX be a smooth and proper rationally connected variety over a number field k k kkk. Let T T TTT be a torus over k k kkk and Y ¯ X k ¯ Y ¯ → X k ¯ bar(Y)rarrX_( bar(k))\bar{Y} \rightarrow X_{\bar{k}}Y¯→Xk¯ a torsor under T k ¯ T k ¯ T_( bar(k))T_{\bar{k}}Tk¯ whose isomorphism class is invariant under Gal ( k ¯ / k ) Gal ⁡ ( k ¯ / k ) Gal( bar(k)//k)\operatorname{Gal}(\bar{k} / k)Gal⁡(k¯/k). Then
X ( A k ) Br ( X ) = f : Y X f ( Y ( A k ) Br ( Y ) ) X A k Br ⁡ ( X ) = ⋃ f : Y → X   f ′ Y ′ A k Br ⁡ Y ′ X(A_(k))^(Br(X))=uuu_(f:Y rarr X)f^(')(Y^(')(A_(k))^(Br(Y^('))))X\left(\mathbf{A}_{k}\right)^{\operatorname{Br}(X)}=\bigcup_{f: Y \rightarrow X} f^{\prime}\left(Y^{\prime}\left(\mathbf{A}_{k}\right)^{\operatorname{Br}\left(Y^{\prime}\right)}\right)X(Ak)Br⁡(X)=⋃f:Y→Xf′(Y′(Ak)Br⁡(Y′))
where the union ranges over the torsors f : Y X f : Y → X f:Y rarr Xf: Y \rightarrow Xf:Y→X under T T TTT whose base change to X k ¯ X k ¯ X_( bar(k))X_{\bar{k}}Xk¯ is isomorphic to Y ¯ Y ¯ bar(Y)\bar{Y}Y¯, and Y Y ′ Y^(')Y^{\prime}Y′ denotes a smooth compactification of Y Y YYY such that f f fff extends to a morphism f : Y X f ′ : Y ′ → X f^('):Y^(')rarr Xf^{\prime}: Y^{\prime} \rightarrow Xf′:Y′→X. In particular, if Y ( k ) Y ′ ( k ) Y^(')(k)Y^{\prime}(k)Y′(k) is dense in Y ( A k ) Br ( Y ) Y ′ A k Br ⁡ Y ′ Y^(')(A_(k))^(Br(Y^(')))Y^{\prime}\left(\mathbf{A}_{k}\right)^{\operatorname{Br}\left(Y^{\prime}\right)}Y′(Ak)Br⁡(Y′) for every such f f fff, then X ( k ) X ( k ) X(k)X(k)X(k) is dense in X ( A k ) Br ( X ) X A k Br ⁡ ( X ) X(A_(k))^(Br(X))X\left(\mathbf{A}_{k}\right)^{\operatorname{Br}(X)}X(Ak)Br⁡(X).
(To bridge the gap between Theorem 2.2 and [50, THÉORÈME 2.1], one needs to know that X ( A k ) Br ( X ) X A k Br ⁡ ( X ) ≠ ∅ X(A_(k))^(Br(X))!=O/X\left(\mathbf{A}_{k}\right)^{\operatorname{Br}(X)} \neq \varnothingX(Ak)Br⁡(X)≠∅ implies the existence of at least one f f fff. This goes back to [25] and follows from [102, THEOREM 3.3.1], [25, PROPOSITION 2.2.5], [103, (3.3)].)

2.4. Fibration

The following fibration theorem suffices for the proof of Theorem 2.1. It results from combining a descent with the work of Harari [35] on the fibration method.
Theorem 2.3. Let p : Z B p : Z → B p:Z rarr Bp: Z \rightarrow Bp:Z→B be a dominant morphism between irreducible, smooth, and proper varieties over a number field k k kkk, with rationally connected generic fibre. Assume that
(1) there exist dense open subsets W Z W ⊂ Z W sub ZW \subset ZW⊂Z and Q B Q ⊂ B Q sub BQ \subset BQ⊂B such that Q Q QQQ is a quasitrivial torus over k k kkk and p p ppp induces a smooth morphism W Q W → Q W rarr QW \rightarrow QW→Q with geometrically irreducible fibres;
(2) the morphism p p ppp admits a rational section over k ¯ k ¯ bar(k)\bar{k}k¯;
(3) for all b B ( k ) b ∈ B ( k ) b in B(k)b \in B(k)b∈B(k) in a dense open subset of B B BBB, the set Z b ( k ) Z b ( k ) Z_(b)(k)Z_{b}(k)Zb(k) is dense in Z b ( A k ) Br ( Z b ) Z b A k Br ⁡ Z b Z_(b)(A_(k))^(Br(Z_(b)))Z_{b}\left(\mathbf{A}_{k}\right)^{\operatorname{Br}\left(Z_{b}\right)}Zb(Ak)Br⁡(Zb)
Then Z ( k ) Z ( k ) Z(k)Z(k)Z(k) is dense in Z ( A k ) Br ( Z ) Z A k Br ⁡ ( Z ) Z(A_(k))^(Br(Z))Z\left(\mathbf{A}_{k}\right)^{\operatorname{Br}(Z)}Z(Ak)Br⁡(Z).
The assumptions of Theorem 2.3 imply that B B BBB is k k kkk-rational. Under the condition that B B BBB is k k kkk-rational, the first two assumptions of Theorem 2.3 are expected to be superfluous (even under weaker hypotheses on the generic fibre of p p ppp than rational connectedness, see [48, COROLLARY 9.23 (1)-(2)]), but removing them altogether is a wide-open problem, well connected with analytic number theory (see [48, § 9], [47]). Removing (2) while keeping (1) might be within reach, though:
Question 2.4. In the statement of Theorem 2.3, can one dispense with the assumption that p p ppp admit a rational section over k ¯ k ¯ bar(k)\bar{k}k¯ ?
This would allow one to replace "supersolvable" with "solvable" in the statement of Theorem 2.1. Indeed, in Section 2.2, the cyclicity of the quotient G m / G m 1 G m / G m − 1 G_(m)//G_(m-1)G_{m} / G_{m-1}Gm/Gm−1 plays a rôle only to ensure the existence of a rational section of p p ppp over k ¯ k ¯ bar(k)\bar{k}k¯ (see [50, PROPOSITION 3.3 (II)]).

2.5. An application to Massey products

Theorem 2.1 has concrete applications, over number fields, beyond the inverse Galois problem: for the homogeneous spaces that appear in its statement, it turns the problem of deciding the existence of a rational point into the much more approachable question of deciding the non-vacuity of the Brauer-Manin set. In this way, Theorem 2.1 can be used to confirm, in the case of number fields, the conjecture of Mináč and Tân on the vanishing of Massey products in Galois cohomology (see [49]). Indeed, this conjecturewhich posits that for any field k k kkk, any prime number p p ppp, any integer m 3 m ≥ 3 m >= 3m \geq 3m≥3 and any classes a 1 , , a m H 1 ( k , Z / p Z ) a 1 , … , a m ∈ H 1 ( k , Z / p Z ) a_(1),dots,a_(m)inH^(1)(k,Z//pZ)a_{1}, \ldots, a_{m} \in H^{1}(k, \mathbf{Z} / p \mathbf{Z})a1,…,am∈H1(k,Z/pZ), the m m mmm-fold Massey product of a 1 , , a m a 1 , … , a m a_(1),dots,a_(m)a_{1}, \ldots, a_{m}a1,…,am vanishes if it is defined (see [78,79])—can be reinterpreted, according to Pál and Schlank [83], in terms of the existence of rational points on appropriate homogeneous spaces of S L n S L n SL_(n)\mathrm{SL}_{n}SLn over k k kkk (with n 0 n ≫ 0 n≫0n \gg 0n≫0 ), and it so happens that the geometric stabilisers of these homogeneous spaces are finite and supersolvable.

3. RATIONAL CURVES ON REAL ALGEBRAIC VARIETIES

3.1. A few questions

Let X X XXX be a smooth variety over R R R\mathbf{R}R. The interplay between the topology of the C C ∞ C^(oo)\mathscr{C}^{\infty}C∞ manifold X ( R ) X ( R ) X(R)X(\mathbf{R})X(R) and the geometry of the algebraic variety X X XXX lies at the core of classical
real algebraic geometry. One of the fundamental problems in this area consists in investigating which submanifolds of X ( R ) X ( R ) X(R)X(\mathbf{R})X(R) can be approximated, in the Euclidean topology, by Zariski closed submanifolds. Even for 1-dimensional submanifolds, i.e., disjoint unions of C C ∞ C^(oo)\mathscr{C}^{\infty}C∞ loops, various phenomena-of a topological, Hodge-theoretic, or yet more subtle nature-can obstruct the existence of algebraic approximations (see [4, § 4]). In the case of 1-dimensional submanifolds, however, all known obstructions vanish when X X XXX is rationally connected. One can thus raise the following questions, in which H 1 alg ( X ( R ) , Z / 2 Z ) H 1 alg  ( X ( R ) , Z / 2 Z ) H_(1)^("alg ")(X(R),Z//2Z)H_{1}^{\text {alg }}(X(\mathbf{R}), \mathbf{Z} / 2 \mathbf{Z})H1alg (X(R),Z/2Z) denotes the image of the cycle class map C H 1 ( X ) H 1 ( X ( R ) , Z / 2 Z ) C H 1 ( X ) → H 1 ( X ( R ) , Z / 2 Z ) CH_(1)(X)rarrH_(1)(X(R),Z//2Z)\mathrm{CH}_{1}(X) \rightarrow H_{1}(X(\mathbf{R}), \mathbf{Z} / 2 \mathbf{Z})CH1(X)→H1(X(R),Z/2Z) defined by Borel and Haefliger [9].
Questions 3.1. Let X X XXX be a smooth, proper, rationally connected variety, over R R R\mathbf{R}R.
(1) Can all C C ∞ C^(oo)\mathscr{C}^{\infty}C∞ loops in X ( R ) X ( R ) X(R)X(\mathbf{R})X(R) be approximated, in the Euclidean topology, by real loci of algebraic curves? or even by real loci of rational algebraic curves?
(2) Is H 1 ( X ( R ) , Z / 2 Z ) = H 1 alg ( X ( R ) , Z / 2 Z ) H 1 ( X ( R ) , Z / 2 Z ) = H 1 alg  ( X ( R ) , Z / 2 Z ) H_(1)(X(R),Z//2Z)=H_(1)^("alg ")(X(R),Z//2Z)H_{1}(X(\mathbf{R}), \mathbf{Z} / 2 \mathbf{Z})=H_{1}^{\text {alg }}(X(\mathbf{R}), \mathbf{Z} / 2 \mathbf{Z})H1(X(R),Z/2Z)=H1alg (X(R),Z/2Z) ? Is H 1 ( X ( R ) , Z / 2 Z ) H 1 ( X ( R ) , Z / 2 Z ) H_(1)(X(R),Z//2Z)H_{1}(X(\mathbf{R}), \mathbf{Z} / 2 \mathbf{Z})H1(X(R),Z/2Z) generated by classes of rational algebraic curves on X X XXX ?
The first parts of Questions 3.1 (1) and (2) are in fact equivalent to each other, by the work of Akbulut and King (see [5, тHEOREM 6.8]), and were studied in a systematic fashion in [4,5]. The second part of Question 3.1 (1) is, however, as far as we know, genuinely stronger than the second part of Question 3.1 (2). We note that in order to formulate the second part of Question 3.1 (1) precisely, it is better to work with possibly noninjective C C ∞ C^(oo)\mathscr{C}^{\infty}C∞ maps P 1 ( R ) X ( R ) P 1 ( R ) → X ( R ) P^(1)(R)rarr X(R)\mathbf{P}^{1}(\mathbf{R}) \rightarrow X(\mathbf{R})P1(R)→X(R) rather than with submanifolds of X ( R ) X ( R ) X(R)X(\mathbf{R})X(R). Indeed, there are examples of R R R\mathbf{R}R-rational surfaces X X XXX and of C C ∞ C^(oo)\mathscr{C}^{\infty}C∞ loops in X ( R ) X ( R ) X(R)X(\mathbf{R})X(R) such that the desired rational algebraic curves necessarily have singular real points (see [68, THEOREM 3]).
A specific motivation for Question 3.1 (2) is its analogy with the following questions in complex geometry raised by Voisin [101] and by Kollár [67]:
Questions 3.2. Let X X XXX be a smooth, proper, rationally connected variety, over C C C\mathbf{C}C. Is the group H 2 ( X ( C ) , Z ) H 2 ( X ( C ) , Z ) H_(2)(X(C),Z)\mathrm{H}_{2}(\mathrm{X}(\mathbf{C}), \mathbf{Z})H2(X(C),Z) generated by homology classes of algebraic curves? Is it generated by homology classes of rational algebraic curves?
The two parts of Questions 3.2 are in fact equivalent: Tian and Zong [100] have shown that the homology class of any algebraic curve on a rationally connected variety over C C C\mathbf{C}C is a linear combination of homology classes of rational curves. The real analogue of their result remains unknown in general. Its validity is an interesting open problem.
The first parts of Questions 3.1 (2) and of Questions 3.2 are in fact related by more than an analogy: if X X XXX is a smooth, proper, rationally connected variety over R R R\mathbf{R}R such that X ( R ) X ( R ) ≠ ∅ X(R)!=O/X(\mathbf{R}) \neq \varnothingX(R)≠∅ and such that Questions 3.2 admit a positive answer for X C X C X_(C)X_{\mathbf{C}}XC, then the equality H 1 ( X ( R ) , Z / 2 Z ) = H 1 alg ( X ( R ) , Z / 2 Z ) H 1 ( X ( R ) , Z / 2 Z ) = H 1 alg  ( X ( R ) , Z / 2 Z ) H_(1)(X(R),Z//2Z)=H_(1)^("alg ")(X(R),Z//2Z)H_{1}(X(\mathbf{R}), \mathbf{Z} / 2 \mathbf{Z})=H_{1}^{\text {alg }}(X(\mathbf{R}), \mathbf{Z} / 2 \mathbf{Z})H1(X(R),Z/2Z)=H1alg (X(R),Z/2Z) is equivalent to the real integral Hodge conjecture for 1-cycles on X X XXX, a property formulated and studied in [4,5].
In a different line of investigation around the abundance of rational curves on rationally connected varieties, many authors have considered the problem of finding rational curves through a prescribed set of points, or more generally through a prescribed curvilinear
0-dimensional subscheme, on any smooth, proper, rationally connected variety X X XXX. Over the complex numbers, such curves exist unconditionally (Kollár, Miyaoka, Mori, see [63, chAPTER IV.3]). Over the real numbers, such curves exist under the necessary condition that all the prescribed points that are real belong to the same connected component of X ( R ) X ( R ) X(R)X(\mathbf{R})X(R) (Kollár, see [ 64 , 66 ] ) [ 64 , 66 ] ) [64,66])[64,66])[64,66]). This problem can be generalised to one-parameter families: given a morphism f : X B f : X → B f:Xrarr Bf: \mathscr{X} \rightarrow Bf:X→B with rationally connected generic fibre between smooth and proper varieties, where B B BBB is a curve, one looks for sections of f f fff whose restriction to a given 0 -dimensional subscheme of B B BBB is prescribed, thus leading to Questions 3.3 below. For simplicity of notation, in the statement of Question 3.3 (2), this 0 -dimensional subscheme of B B BBB is assumed to be reduced; there is, however, no loss of generality in doing this, since jets of sections can be prescribed at any higher order by replacing X X X\mathscr{X}X with a suitable iterated blow-up (see [52, PROPOSITION 1.4]).
Questions 3.3. Let B B BBB be a smooth, proper, connected curve over a field k 0 k 0 k_(0)k_{0}k0. Let X X X\mathscr{X}X be a smooth, proper variety over k 0 k 0 k_(0)k_{0}k0, endowed with a flat morphism f : X B f : X → B f:Xrarr Bf: \mathscr{X} \rightarrow Bf:X→B with rationally connected generic fibre. Let P B P ⊂ B P sub BP \subset BP⊂B be a reduced 0-dimensional subscheme. Let s : P X s : P → X s:P rarrXs: P \rightarrow \mathscr{X}s:P→X be a section of f f fff over P P PPP.
(1) If k 0 = C k 0 = C k_(0)=Ck_{0}=\mathbf{C}k0=C, can s s sss be extended to a section of f f fff ?
(2) If k 0 = R k 0 = R k_(0)=Rk_{0}=\mathbf{R}k0=R and the map s | P ( R ) : P ( R ) X ( R ) s P ( R ) : P ( R ) → X ( R ) s|_(P(R)):P(R)rarrX(R)\left.s\right|_{P(\mathbf{R})}: P(\mathbf{R}) \rightarrow \mathscr{X}(\mathbf{R})s|P(R):P(R)→X(R) can be extended to a C C ∞ C^(oo)\mathscr{C}^{\infty}C∞ section of f | X ( R ) : X ( R ) B ( R ) f X ( R ) : X ( R ) → B ( R ) f|_(X(R)):X(R)rarr B(R)\left.f\right|_{\mathscr{X}(\mathbf{R})}: \mathscr{X}(\mathbf{R}) \rightarrow B(\mathbf{R})f|X(R):X(R)→B(R), can then s s sss be extended to a section of f f fff ?
Let X X XXX be the generic fibre of f f fff and k k kkk the function field of B B BBB. The existence of sections extending any given s s sss as above is equivalent to the density of X ( k ) X ( k ) X(k)X(k)X(k) in the topological space X ( A k ) = b X ( k b ) X A k = ∏ b   X k b X(A_(k))=prod_(b)X(k_(b))X\left(\mathbf{A}_{k}\right)=\prod_{b} X\left(k_{b}\right)X(Ak)=∏bX(kb) of adelic points of X X XXX, where the product runs over the closed points b b bbb of B B BBB and k b k b k_(b)k_{b}kb denotes the completion of k k kkk at b b bbb. This is the weak approximation property.
The Graber-Harris-Starr theorem [34] provides a positive answer to Question 3.3 (1) when P = P = ∅ P=O/P=\varnothingP=∅ and it is a conjecture of Hassett and Tschinkel that the answer to this question is in the affirmative in general (see [ 17 , 52 , 53 , 99 ] [ 17 , 52 , 53 , 99 ] [17,52,53,99][17,52,53,99][17,52,53,99] for known results). Particular cases of Question 3.3 (2) were first studied by Colliot-Thélène [14], who conjectured the validity of weak approximation (i.e., a positive answer to Question 3.3 (2) even without assuming that s | P ( R ) s P ( R ) s|_(P(R))\left.s\right|_{P(\mathbf{R})}s|P(R) can be extended to a C C ∞ C^(oo)\mathscr{C}^{\infty}C∞ section of f | X ( R ) ) f X ( R ) {:f|_(X(R)))\left.\left.f\right|_{\mathscr{X}(\mathbf{R})}\right)f|X(R)) when X X XXX is birationally equivalent to a homogeneous space of a connected linear algebraic group over k k kkk, and proved his conjecture when the geometric stabilisers are trivial. Scheiderer [94] then proved the same conjecture when the geometric stabilisers are connected. Ducros [30,31] stated Question 3.3 (2) in these exact terms, and gave a positive answer when X X XXX is a conic bundle surface, or more generally when there exists a dominant map X P k 1 X → P k 1 X rarrP_(k)^(1)X \rightarrow \mathbf{P}_{k}^{1}X→Pk1 whose generic fibre is a Severi-Brauer variety.

3.2. Tight approximation

The main insight behind the proof of Theorem B is the observation that formulating a suitable common strengthening of Questions 3.1 and 3.3, through the notion of tight approximation, can render all of these questions fully amenable to both the descent method
and the fibration method. We note that Questions 3.1 and Questions 3.3 are somewhat orthogonal in spirit, insofar as the former consider global constraints on curves lying on X X XXX, while the latter are aimed at local constraints.
The idea of establishing a descent method (resp. fibration method) for Question 3.3 (2) already appeared in [31] (resp. [84]), though in [31] and [84] the implementations are subject to miscellaneous restrictions. The possibility of a descent method and a fibration method for studying Questions 3.1, however, is new and turns out to require a shift in perspective from single rationally connected varieties to one-parameter families of such.
Let us illustrate how Questions 3.1 need to be strengthened for a fibration argument to go through. We start with a dominant morphism p : X Y p : X → Y p:X rarr Yp: X \rightarrow Yp:X→Y with rationally connected generic fibre between smooth, proper, rationally connected varieties, over R R R\mathbf{R}R, and a C C ∞ C^(oo)\mathscr{C}{ }^{\infty}C∞ loop γ : S 1 X ( R ) γ : S 1 → X ( R ) gamma:S^(1)rarr X(R)\gamma: \mathbf{S}^{1} \rightarrow X(\mathbf{R})γ:S1→X(R) that we want to approximate, in the Euclidean topology, by a Zariski closed submanifold of X X XXX, assuming that we can solve the same problem on Y Y YYY as well as on the fibres of p p ppp. By assumption, we can approximate p γ : S 1 Y ( R ) p ∘ γ : S 1 → Y ( R ) p@gamma:S^(1)rarr Y(R)p \circ \gamma: \mathbf{S}^{1} \rightarrow Y(\mathbf{R})p∘γ:S1→Y(R) by a C C ∞ C^(oo)\mathscr{C}^{\infty}C∞ map ξ : S 1 Y ( R ) ξ : S 1 → Y ( R ) xi:S^(1)rarr Y(R)\xi: \mathbf{S}^{1} \rightarrow Y(\mathbf{R})ξ:S1→Y(R) with Zariski closed image. The best we can hope to find, then, is a C C ∞ C^(oo)\mathscr{C}{ }^{\infty}C∞ loop γ ~ : S 1 X ( R ) γ ~ : S 1 → X ( R ) tilde(gamma):S^(1)rarr X(R)\tilde{\gamma}: \mathbf{S}^{1} \rightarrow X(\mathbf{R})γ~:S1→X(R) arbitrarily close to γ γ gamma\gammaγ and such that p γ ~ = ξ p ∘ γ ~ = ξ p@ tilde(gamma)=xip \circ \tilde{\gamma}=\xip∘γ~=ξ. We draw two conclusions:
(1) If such a γ ~ γ ~ tilde(gamma)\tilde{\gamma}γ~ exists, the next and final step is not finding an algebraic approximation for a C C ∞ C^(oo)\mathscr{C}^{\infty}C∞ loop in a fibre of p p ppp, but, rather, considering the algebraic curve B B BBB underlying ξ ( S 1 ) ξ S 1 xi(S^(1))\xi\left(\mathbf{S}^{1}\right)ξ(S1), viewing γ ~ γ ~ tilde(gamma)\tilde{\gamma}γ~ as a C C ∞ C^(oo)\mathscr{C}^{\infty}C∞ section of the projection
( X × Y B ) ( R ) B ( R ) X × Y B ( R ) → B ( R ) (Xxx_(Y)B)(R)rarr B(R)\left(X \times_{Y} B\right)(\mathbf{R}) \rightarrow B(\mathbf{R})(X×YB)(R)→B(R)
and looking for an algebraic section of X × Y B B X × Y B → B Xxx_(Y)B rarr BX \times_{Y} B \rightarrow BX×YB→B approximating γ ~ γ ~ tilde(gamma)\tilde{\gamma}γ~. Thus, even when we start with just two real varieties X X XXX and Y Y YYY, we need to consider one-parameter algebraic families of fibres of p p ppp, rather than single fibres.
(2) Consider the example where p p ppp is the blow-up of a surface Y Y YYY at a real point b b bbb and γ γ gamma\gammaγ meets p 1 ( b ) ( R ) p − 1 ( b ) ( R ) p^(-1)(b)(R)p^{-1}(b)(\mathbf{R})p−1(b)(R), transversally. Then for any γ ~ γ ~ tilde(gamma)\tilde{\gamma}γ~ sufficiently close to γ γ gamma\gammaγ in the Euclidean topology, the loop p γ ~ p ∘ γ ~ p@ tilde(gamma)p \circ \tilde{\gamma}p∘γ~ has to go through b b bbb. Hence ξ ξ xi\xiξ has to be required to go through b b bbb for a loop γ ~ γ ~ tilde(gamma)\tilde{\gamma}γ~ as above to exist. Thus, a condition of weak approximation type must be considered in conjunction with Questions 3.1 (as was already noted by Bochnak and Kucharz [8]).
Let us now similarly contemplate a fibration argument in the context of Question 3.3 (2). We assume that X f B X → f B Xrarr"f"B\mathscr{X} \xrightarrow{f} BX→fB can be factored as X p Y g B X → p Y → g B Xrarr"p"Yrarr"g"B\mathscr{X} \xrightarrow{p} \mathscr{Y} \xrightarrow{g} BX→pY→gB, where the variety Y Y Y\mathscr{Y}Y is smooth and proper over R R R\mathbf{R}R, the morphism p p ppp is dominant with rationally connected generic fibre, and g g ggg is flat. Starting from a section s : P X s : P → X s:P rarrXs: P \rightarrow \mathscr{X}s:P→X of f f fff over P P PPP such that s | P ( R ) s P ( R ) s|_(P(R))\left.s\right|_{P(\mathbf{R})}s|P(R) can be extended to a C C ∞ C^(oo)\mathscr{C}^{\infty}C∞ section s s ′ s^(')s^{\prime}s′ of f | X ( R ) f X ( R ) f|_(X(R))\left.f\right|_{\mathscr{X}(\mathbf{R})}f|X(R), a positive answer to Question 3.3 (2) for g g ggg produces for us a section τ Ï„ tau\tauÏ„ of g g ggg that extends p s p ∘ s p@sp \circ sp∘s. Let Z = p 1 ( τ ( B ) ) Z = p − 1 ( Ï„ ( B ) ) Z=p^(-1)(tau(B))\mathscr{Z}=p^{-1}(\tau(B))Z=p−1(Ï„(B)) and let h : Z B h : Z → B h:Zrarr Bh: \mathscr{Z} \rightarrow Bh:Z→B denote the restriction of f f fff. At this point, one would like to apply a positive answer to Question 3.3 (2) for h h hhh to obtain a section of h h hhh extending s s sss, thus completing the argument, as Z X Z ⊆ X ZsubeX\mathscr{Z} \subseteq \mathscr{X}Z⊆X. In order to do so, one needs to know that s | P ( R ) : P ( R ) Z ( R ) s P ( R ) : P ( R ) → Z ( R ) s|_(P(R)):P(R)rarrZ(R)\left.s\right|_{P(\mathbf{R})}: P(\mathbf{R}) \rightarrow \mathscr{Z}(\mathbf{R})s|P(R):P(R)→Z(R) can be extended to a C C ∞ C^(oo)\mathscr{C}^{\infty}C∞ section of h | Z ( R ) : Z ( R ) B ( R ) h Z ( R ) : Z ( R ) → B ( R ) h|_(Z(R)):Z(R)rarr B(R)\left.h\right|_{\mathscr{Z}(\mathbf{R})}: \mathscr{Z}(\mathbf{R}) \rightarrow B(\mathbf{R})h|Z(R):Z(R)→B(R). However, the map h | Z ( R ) h Z ( R ) h|_(Z(R))\left.h\right|_{\mathscr{Z}(\mathbf{R})}h|Z(R) in general even fails
to be surjective. To correct this problem, one should require, at the very least, that τ ( B ( R ) ) Ï„ ( B ( R ) ) tau(B(R))\tau(B(\mathbf{R}))Ï„(B(R)) approximate, in the Euclidean topology, the image of p s : B ( R ) Y ( R ) p ∘ s ′ : B ( R ) → Y ( R ) p@s^('):B(R)rarrY(R)p \circ s^{\prime}: B(\mathbf{R}) \rightarrow \mathscr{Y}(\mathbf{R})p∘s′:B(R)→Y(R). Thus, all in all, an approximation condition in the Euclidean topology has to be considered in conjunction with Question 3.3 (2).
The above discussion leads to the following definition. (This definition slightly differs from that given in [6], which considers the more general question of approximating holomorphic maps by algebraic ones, à la Runge, and which, as a consequence, is useful also for studying complex curves on complex varieties, without reference to the reals; however, all of the statements we make below are true with respect to either of the definitions.)
Definition 3.4. Let B B BBB be a smooth, proper, connected curve over R. A variety X X XXX over k = R ( B ) k = R ( B ) k=R(B)k=\mathbf{R}(B)k=R(B) satisfies the tight approximation property if for any proper model f : X B f : X → B f:Xrarr Bf: \mathscr{X} \rightarrow Bf:X→B of X X XXX over B B BBB with X X X\mathscr{X}X smooth over R R R\mathbf{R}R, any reduced 0 -dimensional subscheme P B P ⊂ B P sub BP \subset BP⊂B, any section s : P X s ′ : P → X s^('):P rarrXs^{\prime}: P \rightarrow \mathscr{X}s′:P→X of f f fff over P P PPP and any C C ∞ C^(oo)\mathscr{C}^{\infty}C∞ section s : B ( R ) X ( R ) s : B ( R ) → X ( R ) s:B(R)rarrX(R)s: B(\mathbf{R}) \rightarrow \mathscr{X}(\mathbf{R})s:B(R)→X(R) of f | X ( R ) f X ( R ) f|_(X(R))\left.f\right|_{\mathscr{X}(\mathbf{R})}f|X(R) such that s | P ( R ) = s | P ( R ) s P ( R ) = s ′ P ( R ) s|_(P(R))=s^(')|_(P(R))\left.s\right|_{P(\mathbf{R})}=\left.s^{\prime}\right|_{P(\mathbf{R})}s|P(R)=s′|P(R), there exists a section σ : B X σ : B → X sigma:B rarrX\sigma: B \rightarrow \mathscr{X}σ:B→X of f f fff such that σ | P = s | P σ P = s ′ P sigma|_(P)=s^(')|_(P)\left.\sigma\right|_{P}=\left.s^{\prime}\right|_{P}σ|P=s′|P and such that σ | B ( R ) σ B ( R ) sigma|_(B(R))\left.\sigma\right|_{B(\mathbf{R})}σ|B(R) lies arbitrarily close to s s sss in the compact-open topology.
Given a smooth, proper, rationally connected variety X X XXX over R R R\mathbf{R}R, the validity of the tight approximation property for the variety obtained from X X XXX by extension of scalars from R R R\mathbf{R}R to R ( t ) R ( t ) R(t)\mathbf{R}(t)R(t) implies positive answers to Questions 3.1 for X X XXX.
The tight approximation property is (tautologically) a birational invariant, and it holds for P k n P k n P_(k)^(n)\mathbf{P}_{k}^{n}Pkn by a theorem of Bochnak and Kucharz [8]. (In [8], weak approximation conditions at complex points are ignored, but they create no additional difficulty.) The next two results provide more examples of varieties satisfying tight approximation.

3.3. Descent

The following theorem implements the descent method for the tight approximation property, in full generality (including non-abelian descent, as formalised by Harari and Skorobogatov). Its proof, given in [6], builds on the work of Scheiderer [94] and, in the case where G G GGG is finite, on an argument of Colliot-Thélène and Gille [17].
Theorem 3.5. Let k k kkk be the function field of a real curve. Let X X XXX be a smooth variety over k k kkk. Let G G GGG be a linear algebraic group over k k kkk. Let f : Y X f : Y → X f:Y rarr Xf: Y \rightarrow Xf:Y→X be a left torsor under G G GGG. Consider twists f : Y X f ′ : Y ′ → X f^('):Y^(')rarr Xf^{\prime}: Y^{\prime} \rightarrow Xf′:Y′→X of f f fff by right torsors under G G GGG, over k k kkk. If every such Y Y ′ Y^(')Y^{\prime}Y′ satisfies the tight approximation property, then so does X X XXX.

3.4. Fibration

The next theorem implements the fibration method for the tight approximation property, in full generality. Its proof, contained in [6], makes essential use of the weak toroidalisation theorem of Abramovich, Denef, and Karu [1] to establish a version of the Néron smoothening process (as in [ 11 , 3.1 / 3 ] [ 11 , 3.1 / 3 ] [11,3.1//3][11,3.1 / 3][11,3.1/3] ) for higher-dimensional bases-the point being that in the discussion at the beginning of Section 3.2, the loop γ ~ γ ~ tilde(gamma)\tilde{\gamma}γ~ is easily seen to exist once the morphism p p ppp is smooth along γ γ gamma\gammaγ (see [5, LEMMA 6.11]).
Theorem 3.6. Let k k kkk be the function field of a real curve. Let p : Z B p : Z → B p:Z rarr Bp: Z \rightarrow Bp:Z→B be a dominant morphism between smooth varieties over k k kkk. If B B BBB and the fibres of p p ppp above the rational points of a dense open subset of B satisfy the tight approximation property, then so does Z Z ZZZ.

3.5. Homogeneous spaces

We are now in a position to sketch the proof of the following theorem, which in the "constant case," i.e., when the algebraic group and the homogeneous space are both defined over R R R\mathbf{R}R, immediately implies Theorem B.
Theorem 3.7. Homogeneous spaces of connected linear algebraic groups over the function field of a real curve satisfy the tight approximation property.
The proof of Theorem 3.7 starts by noting that quasitrivial tori over k k kkk are k k kkk-rational, hence satisfy the tight approximation property (since so does P k n P k n P_(k)^(n)\mathbf{P}_{k}^{n}Pkn ). Any torus T T TTT can be inserted into an exact sequence 1 S Q T 1 1 → S → Q → T → 1 1rarr S rarr Q rarr T rarr11 \rightarrow S \rightarrow Q \rightarrow T \rightarrow 11→S→Q→T→1 where S S SSS is a torus and Q Q QQQ is a quasitrivial torus. As any twist of Q Q QQQ as a torsor remains isomorphic to Q Q QQQ (Hilbert's Theorem 90) and hence satisfies the tight approximation property, we deduce, by the descent method (Theorem 3.5), that all tori over k k kkk satisfy the tight approximation property. Next, as every connected linear algebraic group over k k kkk is birationally equivalent to a relative torus over a k k kkk-rational variety (namely over the variety of maximal tori, when the algebraic group is reductive), we deduce, by the fibration method (Theorem 3.6), that connected linear algebraic groups over k k kkk satisfy the tight approximation property. By descent (Theorem 3.5 again), it follows that homogeneous spaces of connected linear algebraic groups over k k kkk satisfy the tight approximation property when they have a rational point. Finally, it is a theorem of Scheiderer that homogeneous spaces of connected linear algebraic groups over k k kkk satisfy the Hasse principle with respect to the real closures of k k kkk, so that if X X XXX denotes such a homogeneous space, then X ( k ) X ( k ) ≠ ∅ X(k)!=O/X(k) \neq \varnothingX(k)≠∅ whenever a C C ∞ C^(oo)\mathscr{C}^{\infty}C∞ section s : B ( R ) X ( R ) s : B ( R ) → X ( R ) s:B(R)rarrX(R)s: B(\mathbf{R}) \rightarrow \mathscr{X}(\mathbf{R})s:B(R)→X(R) as in Definition 3.4 exists. This completes the proof of Theorem 3.7.

3.6. Further comments

Theorem 3.7 implies that homogeneous spaces of connected linear algebraic groups over the function field of a real curve satisfy weak approximation, as conjectured by ColliotThélène. Indeed, in the notation of Definition 3.4, if X X XXX is such a homogeneous space and P P PPP contains the locus of singular fibres of f f fff, Scheiderer's work implies that f 1 ( b ) ( R ) f − 1 ( b ) ( R ) f^(-1)(b)(R)f^{-1}(b)(\mathbf{R})f−1(b)(R) is nonempty and connected for all b B ( R ) P ( R ) b ∈ B ( R ) ∖ P ( R ) b in B(R)\\P(R)b \in B(\mathbf{R}) \backslash P(\mathbf{R})b∈B(R)∖P(R), so that a C C ∞ C^(oo)\mathscr{C}^{\infty}C∞ section s : B ( R ) X ( R ) s : B ( R ) → X ( R ) s:B(R)rarrX(R)s: B(\mathbf{R}) \rightarrow \mathscr{X}(\mathbf{R})s:B(R)→X(R) with s | P ( R ) = s | P ( R ) s P ( R ) = s ′ P ( R ) s|_(P(R))=s^(')|_(P(R))\left.s\right|_{P(\mathbf{R})}=\left.s^{\prime}\right|_{P(\mathbf{R})}s|P(R)=s′|P(R) always exists.
The main open problem surrounding the notion of tight approximation is the following.
Question 3.8. Let k k kkk be the function field of a real curve. Do all rationally connected varieties over k k kkk satisfy the tight approximation property?
Building on Theorems 3.5 and 3.6, the tight approximation property is shown in [6] to hold for various classes of rationally connected varieties beyond homogeneous spaces of connected linear algebraic groups. For instance, it holds for smooth cubic hypersurfaces
of dimension 2 ≥ 2 >= 2\geq 2≥2 that are defined over R R R\mathbf{R}R, thus yielding, for such hypersurfaces, a positive answer to (the second part of) Question 3.1 (1).
Question 3.8 is open for cubic surfaces over k k kkk. Even Question 3.3 (2) is open when X X XXX is a cubic surface, although Question 3.3 (1) has an affirmative answer in this case, by a theorem of Tian [99].
In another direction, Question 3.8 is open for surfaces defined over R R R\mathbf{R}R, and so is (the second part of) Question 3.1 (1). By inspecting the birational classification of geometrically rational surfaces and using the fibration method (Theorem 3.6), one can see that a positive answer to these questions for surfaces defined over R R R\mathbf{R}R would follow from a positive answer for del Pezzo surfaces of degree 1 or 2 defined over R R R\mathbf{R}R. In these cases, it would suffice, by an application of the descent method (Theorem 3.5), to know that for any real del Pezzo surface X X XXX of degree 1 or 2 , the universal torsors of X X XXX, in the sense of Colliot-Thélène and Sansuc [25], are R-rational whenever they have a real point. This last question, unfortunately, is very much open-even the unirationality of real del Pezzo surfaces of degree 1 is unknown. In fact, not a single example of a minimal real del Pezzo surface of degree 1 is known to be unirational. For a description of these surfaces, see [93, & 5].
Naturally, one hopes for the answer to Question 3.8 to be in the affirmative in general. This conjecture would have a host of interesting consequences, among which: a version of the Graber-Harris-Starr theorem over the reals (i.e., a positive answer to Question 3.3 (2) when P = P = ∅ P=O/P=\varnothingP=∅ ); Lang's widely open conjecture from [70] that the function field of a real curve with no real point is C 1 C 1 C_(1)C_{1}C1 (see [55, coRoLLARY 1.5] for the implication); and the existence of a geometrically rational curve on any smooth, proper, rationally connected variety of dimension 1 ≥ 1 >= 1\geq 1≥1 over R R R\mathbf{R}R.
This last consequence is a conjecture of Kollár, who showed the existence of rational curves on those real rationally connected varieties of dimension 1 ≥ 1 >= 1\geq 1≥1 that have real points (see [2, Remarks 20]). For real rationally connected varieties with no real point, it is interesting to consider a weaker property: the existence of a geometrically irreducible curve of even geometric genus. The latter can be reinterpreted in terms of the real integral Hodge conjecture (see [4]). Using Hodge theory and a real adaptation of Green's infinitesimal criterion for the density of Noether-Lefschetz loci, such curves of even genus can be shown to exist on all real Fano threefolds (see [5]). However, even on smooth quartic hypersurfaces in P R 4 P R 4 P_(R)^(4)\mathbf{P}_{\mathbf{R}}^{4}PR4, the existence of geometrically rational curves remains a challenge, as well as the mere existence of an absolute bound, independent of the chosen quartic hypersurface, on the minimal geometric genus of a geometrically irreducible curve of even geometric genus lying on such a hypersurface.

4. FUNCTION FIELDS OF CURVES OVER p p ppp-ADIC FIELDS

4.1. Some motivation: rational curves over number fields

Even though the main questions about rational points of rationally connected varieties over number fields and over function fields of real curves are still wide open, the
Brauer-Manin obstruction and the tight approximation property at least provide rather satisfactory conjectural answers. It would be highly desirable to obtain a similar conjectural picture for rational points over other fields, for significant classes of varieties-including, at a minimum, concrete criteria for the existence of rational points.
Over the field Q ( t ) Q ( t ) Q(t)\mathbf{Q}(t)Q(t), this would encompass questions about rational curves on rationally connected varieties over Q Q Q\mathbf{Q}Q, about which very little is known. For example, it is unknown whether any rationally connected variety of dimension 1 ≥ 1 >= 1\geq 1≥1 over Q Q Q\mathbf{Q}Q that possesses a rational point also contains a rational curve defined over Q Q Q\mathbf{Q}Q. Much more ambitiously, it is unknown whether any such variety contains enough rational curves to imply the finiteness of the set of R R RRR-equivalence classes of rational points, a question asked in [16, QuEsTION 10.12]. (Known results on this problem are listed after Question 10.12 in [16].) As another example, the regular inverse Galois problem over Q Q Q\mathbf{Q}Q, which asks for the construction of a regular Galois extension of Q ( t ) Q ( t ) Q(t)\mathbf{Q}(t)Q(t) with specified Galois group, and which can be reinterpreted as a problem about the existence of appropriate rational curves on the homogeneous space S L n / G S L n / G SL_(n)//G\mathrm{SL}_{n} / GSLn/G over Q Q Q\mathbf{Q}Q is open even for finite nilpotent groups G G GGG. All of these problems are currently out of reach.
As a first step towards these questions, let us replace Q Q Q\mathbf{Q}Q with its completions and turn to rational points over the field Q p ( t ) Q p ( t ) Q_(p)(t)\mathbf{Q}_{p}(t)Qp(t) or over its finite extensions.

4.2. Rational curves on varieties over p p ppp-adic fields

In the constant case (that is, for varieties obtained by scalar extension from varieties defined over a p p ppp-adic field, i.e., a finite extension of Q p Q p Q_(p)\mathbf{Q}_{p}Qp ), various existence results are known:
(1) the regular inverse Galois problem over Q p Q p Q_(p)\mathbf{Q}_{p}Qp has a positive solution (first proved by Harbater [39], by "formal patching"; reproved and generalised in different directions by Pop [91] and by Colliot-Thélène [15]; see also [65,74,80]);
(2) for any smooth, proper, rationally connected variety X X XXX over a p p ppp-adic field k k kkk, Kollár [ 64 , 66 ] [ 64 , 66 ] [64,66][64,66][64,66] has shown that the rational points of X X XXX fall into finitely many R R RRR-equivalence classes, and that there exist rational curves on X X XXX, defined over k k kkk, passing through any finite set of rational points of X X XXX that belong to the same R R RRR-equivalence class (with prescribed jets of any given order at these points).
This last statement concerns conditions of weak approximation type that can be imposed on rational curves on rationally connected varieties over p p ppp-adic fields. It would be interesting to formulate an analogue, in this p p ppp-adic context, of the surjectivity of the Borel-Haefliger cycle class map C H 1 ( X ) H 1 ( X ( R ) , Z / 2 Z ) C H 1 ( X ) → H 1 ( X ( R ) , Z / 2 Z ) CH_(1)(X)rarrH_(1)(X(R),Z//2Z)\mathrm{CH}_{1}(X) \rightarrow H_{1}(X(\mathbf{R}), \mathbf{Z} / 2 \mathbf{Z})CH1(X)→H1(X(R),Z/2Z) (i.e., of Questions 3.1 (2)).
We saw in Section 3 that in order to answer questions about homology classes of rational curves on real varieties, it can be useful to consider more generally the tight approximation property, for nonconstant varieties over the function field of a real curve. By analogy, this gives incentive to investigate the possibility of a p p ppp-adic analogue of the tight approximation property for nonconstant varieties over the function field of a curve over a p p ppp-adic field, the validity of which would have consequences for a likely easier to formulate p p ppp-adic integral Hodge conjecture for 1-cycles on varieties over p p ppp-adic fields.

4.3. Quadrics and other homogeneous spaces

In the nonconstant case, even the simplest varieties over Q p ( t ) Q p ( t ) Q_(p)(t)\mathbf{Q}_{p}(t)Qp(t) lead to difficult problems when it comes to their rational points. For instance, it is only a relatively recent theorem of Parimala and Suresh [86], for p 2 p ≠ 2 p!=2p \neq 2p≠2, and of Leep [72], based on work of Heath-Brown [54], for arbitrary p p ppp, that every projective quadric of dimension 7 ≥ 7 >= 7\geq 7≥7 over Q p ( t ) Q p ( t ) Q_(p)(t)\mathbf{Q}_{p}(t)Qp(t) possesses a rational point. (In the language of quadratic forms, "the u u uuu-invariant of Q p ( t ) Q p ( t ) Q_(p)(t)\mathbf{Q}_{p}(t)Qp(t) is equal to 8.") Many other articles have been devoted to local-global principles for varieties over function fields of curves over p p ppp-adic fields (e.g., [19-21, 23, 24,37,38,40-46,56-58,77,85,87,88,92,98]).
A patching technique was developed by Harbater, Hartmann and Krashen ("patching over fields," a successor to formal patching), and was applied to study rational points of homogeneous spaces over such fields. It was used, in [42], to give another proof of the aforementioned theorem of Parimala and Suresh, and, in [23], to establish, more generally, the local-global principle for the existence of rational points on smooth projective quadrics of dimension 1 ≥ 1 >= 1\geq 1≥1 over Q p ( t ) Q p ( t ) Q_(p)(t)\mathbf{Q}_{p}(t)Qp(t) (or over a finite extension of Q p ( t ) Q p ( t ) Q_(p)(t)\mathbf{Q}_{p}(t)Qp(t) ), with respect to all discrete valuations on this field, when p p ppp is odd.

4.4. Reciprocity obstructions

Let k k kkk be a finite extension of Q p ( t ) Q p ( t ) Q_(p)(t)\mathbf{Q}_{p}(t)Qp(t). Let Ω Î© Omega\OmegaΩ denote the set of equivalence classes of discrete valuations (of rank 1) on k k kkk and, for v Ω v ∈ Ω v in Omegav \in \Omegav∈Ω, let k v k v k_(v)k_{v}kv denote the completion of k k kkk at v v vvv. Let X X XXX be an irreducible, smooth and proper variety over k k kkk. We embed X ( k ) X ( k ) X(k)X(k)X(k) diagonally into the product topological space v Ω X ( k v ) ∏ v ∈ Ω   X k v prod_(v in Omega)X(k_(v))\prod_{v \in \Omega} X\left(k_{v}\right)∏v∈ΩX(kv), which we shall also denote X ( A k ) X A k X(A_(k))X\left(\mathbf{A}_{k}\right)X(Ak) (recall that X X XXX is proper).
We now explain how, building on the work of Bloch-Ogus and of Kato, an analogue of the Brauer-Manin obstruction can be set up in this context. These ideas, which are due to Colliot-Thélène, appear in print, and are put to use, in [ 24 , 82.3 ] [ 24 , 82.3 ] [24,82.3][24,82.3][24,82.3], in a very slightly different (equicharacteristic) situation. We refer the reader to [24, § 2.3] for more details. (The "reciprocity obstructions" of [ 37 , § 4 ] [ 37 , § 4 ] [37,§4][37, \S 4][37,§4] are weaker than those we discuss here.)
Our goal is thus to define, in complete generality, a closed subset X ( A k ) r e c X ( A k ) X A k r e c ⊆ X A k X(A_(k))^(rec)sube X(A_(k))X\left(\mathbf{A}_{k}\right)^{\mathrm{rec}} \subseteq X\left(\mathbf{A}_{k}\right)X(Ak)rec⊆X(Ak) containing X ( k ) X ( k ) X(k)X(k)X(k), using on the one hand a reciprocity law coming from k k kkk and on the other hand an analogue of the Brauer group of X X XXX.
Grothendieck's purity theorem for the Brauer group equates Br ( X ) Br ⁡ ( X ) Br(X)\operatorname{Br}(X)Br⁡(X) with the unramified cohomology group H n r 2 ( X / k , Q / Z ( 1 ) ) H n r 2 ( X / k , Q / Z ( 1 ) ) H_(nr)^(2)(X//k,Q//Z(1))H_{\mathrm{nr}}^{2}(X / k, \mathbf{Q} / \mathbf{Z}(1))Hnr2(X/k,Q/Z(1)). We recall the definition of unramified cohomology: for any irreducible smooth variety V V VVV over a field K K KKK of characteristic 0 and any torsion Galois module M M MMM over K K KKK, the group H n r q ( V / K , M ) H n r q ( V / K , M ) H_(nr)^(q)(V//K,M)H_{\mathrm{nr}}^{q}(V / K, M)Hnrq(V/K,M) is the subgroup of the Galois cohomology group H q ( K ( V ) , M ) H q ( K ( V ) , M ) H^(q)(K(V),M)H^{q}(K(V), M)Hq(K(V),M) consisting of those classes whose residues along all codimension 1 points of V V VVV vanish. It is the unramified cohomology group H n r 3 ( X / k , Q / Z ( 2 ) ) H n r 3 ( X / k , Q / Z ( 2 ) ) H_(nr)^(3)(X//k,Q//Z(2))H_{\mathrm{nr}}^{3}(X / k, \mathbf{Q} / \mathbf{Z}(2))Hnr3(X/k,Q/Z(2)) that will serve as a substitute for Br ( X ) Br ⁡ ( X ) Br(X)\operatorname{Br}(X)Br⁡(X) here. (The shift in degree is explained by the fact that the field k k kkk has cohomological dimension 3 while number fields have virtual cohomological dimension 2.) For any field extension K / k K / k K//kK / kK/k, Bloch-Ogus theory provides an evaluation map H n r 3 ( X / k , Q / Z ( 2 ) ) H 3 ( K , Q / Z ( 2 ) ) , α α ( x ) H n r 3 ( X / k , Q / Z ( 2 ) ) → H 3 ( K , Q / Z ( 2 ) ) , α ↦ α ( x ) H_(nr)^(3)(X//k,Q//Z(2))rarrH^(3)(K,Q//Z(2)),alpha|->alpha(x)H_{\mathrm{nr}}^{3}(X / k, \mathbf{Q} / \mathbf{Z}(2)) \rightarrow H^{3}(K, \mathbf{Q} / \mathbf{Z}(2)), \alpha \mapsto \alpha(x)Hnr3(X/k,Q/Z(2))→H3(K,Q/Z(2)),α↦α(x) along any K K KKK-point x x xxx of X X XXX (see [7]).
Let B B B\mathscr{B}B denote an irreducible normal proper scheme over Z p Z p Z_(p)\mathbf{Z}_{p}Zp with function field k k kkk. In contrast with what happens over number fields, here it is not one reciprocity law that will play a rôle, but infinitely many of them: one for each closed point of B B B\mathscr{B}B, for each such B B B\mathscr{B}B. Namely, given any closed point b B b ∈ B b inBb \in \mathscr{B}b∈B, Kato [ 61 , § 1 ] [ 61 , § 1 ] [61,§1][61, \S 1][61,§1] has constructed a complex
(4.1) H 3 ( k , Q / Z ( 2 ) ) ξ B 1 , b Br ( κ ( ξ ) ) Q / Z (4.1) H 3 ( k , Q / Z ( 2 ) ) → ⨁ ξ ∈ B 1 , b   Br ⁡ ( κ ( ξ ) ) → Q / Z {:(4.1)H^(3)(k","Q//Z(2))rarrbigoplus_(xi inB_(1,b))Br(kappa(xi))rarrQ//Z:}\begin{equation*} H^{3}(k, \mathbf{Q} / \mathbf{Z}(2)) \rightarrow \bigoplus_{\xi \in \mathscr{B}_{1, b}} \operatorname{Br}(\kappa(\xi)) \rightarrow \mathbf{Q} / \mathbf{Z} \tag{4.1} \end{equation*}(4.1)H3(k,Q/Z(2))→⨁ξ∈B1,bBr⁡(κ(ξ))→Q/Z
where ξ ξ xi\xiξ ranges over the set B 1 , b B 1 , b B_(1,b)\mathscr{B}_{1, b}B1,b of 1-dimensional irreducible closed subsets of B B B\mathscr{B}B that contain b b bbb, and where κ ( ξ ) κ ( ξ ) kappa(xi)\kappa(\xi)κ(ξ) denotes the function field of ξ ξ xi\xiξ (which is either a global field of characteristic p p ppp or a local field of characteristic 0 ). The second arrow in (4.1) is the sum of the invariant maps from local class field theory at the finitely many places of κ ( ξ ) κ ( ξ ) kappa(xi)\kappa(\xi)κ(ξ) that lie over b b bbb. The first arrow of (4.1) is induced by residue maps v : H 3 ( k v , Q / Z ( 2 ) ) Br ( κ ( ξ ) ) ∂ v : H 3 k v , Q / Z ( 2 ) → Br ⁡ ( κ ( ξ ) ) del_(v):H^(3)(k_(v),Q//Z(2))rarr Br(kappa(xi))\partial_{v}: H^{3}\left(k_{v}, \mathbf{Q} / \mathbf{Z}(2)\right) \rightarrow \operatorname{Br}(\kappa(\xi))∂v:H3(kv,Q/Z(2))→Br⁡(κ(ξ)) constructed by Kato in [61], where v v vvv denotes the discrete valuation of k k kkk defined by ξ ξ xi\xiξ.
For any α H n r 3 ( X / k , Q / Z ( 2 ) ) α ∈ H n r 3 ( X / k , Q / Z ( 2 ) ) alpha inH_(nr)^(3)(X//k,Q//Z(2))\alpha \in H_{\mathrm{nr}}^{3}(X / k, \mathbf{Q} / \mathbf{Z}(2))α∈Hnr3(X/k,Q/Z(2)), there are only finitely many 1-dimensional irreducible closed subsets ξ ξ xi\xiξ of B B B\mathscr{B}B such that the map X ( k v ) Br ( κ ( ξ ) ) , x v ( α ( x ) ) X k v → Br ⁡ ( κ ( ξ ) ) , x ↦ ∂ v ( α ( x ) ) X(k_(v))rarr Br(kappa(xi)),x|->del_(v)(alpha(x))X\left(k_{v}\right) \rightarrow \operatorname{Br}(\kappa(\xi)), x \mapsto \partial_{v}(\alpha(x))X(kv)→Br⁡(κ(ξ)),x↦∂v(α(x)) does not identically vanish, if we denote by v v vvv the discrete valuation of k k kkk defined by ξ ξ xi\xiξ (see [24, PRoPoSITION 2.7 (II)] and note that for the proof given there, it is enough to assume that a dense open subset of B B B\mathscr{B}B, rather than B B B\mathscr{B}B itself, is a scheme over a field-an assumption satisfied here). As a consequence, it makes sense to define X ( A k ) rec X A k rec  X(A_(k))^("rec ")X\left(\mathbf{A}_{k}\right)^{\text {rec }}X(Ak)rec  to be the set of ( x v ) v Ω X ( A k ) x v v ∈ Ω ∈ X A k (x_(v))_(v in Omega)in X(A_(k))\left(x_{v}\right)_{v \in \Omega} \in X\left(\mathbf{A}_{k}\right)(xv)v∈Ω∈X(Ak) such that for any irreducible normal proper scheme B B B\mathscr{B}B over Z p Z p Z_(p)\mathbf{Z}_{p}Zp with function field k k kkk, for any closed point b B b ∈ B b inBb \in \mathscr{B}b∈B, and for any α H n r 3 ( X / k , Q / Z ( 2 ) ) α ∈ H n r 3 ( X / k , Q / Z ( 2 ) ) alpha inH_(nr)^(3)(X//k,Q//Z(2))\alpha \in H_{\mathrm{nr}}^{3}(X / k, \mathbf{Q} / \mathbf{Z}(2))α∈Hnr3(X/k,Q/Z(2)), the family ( v ( α ( x v ) ) ) ξ B 1 , b ∂ v α x v ξ ∈ B 1 , b ∈ (del_(v)(alpha(x_(v))))_(xi inB_(1,b))in\left(\partial_{v}\left(\alpha\left(x_{v}\right)\right)\right)_{\xi \in \mathscr{B}_{1, b}} \in(∂v(α(xv)))ξ∈B1,b∈ ξ B 1 , b Br ( κ ( ξ ) ) ⨁ ξ ∈ B 1 , b   Br ⁡ ( κ ( ξ ) ) bigoplus_(xi inB_(1,b))Br(kappa(xi))\bigoplus_{\xi \in \mathscr{B}_{1, b}} \operatorname{Br}(\kappa(\xi))⨁ξ∈B1,bBr⁡(κ(ξ)) belongs to the kernel of the second arrow of (4.1). The fact that (4.1) is a complex immediately implies that X ( k ) X ( A k ) rec X ( k ) ⊆ X A k rec  X(k)sube X(A_(k))^("rec ")X(k) \subseteq X\left(\mathbf{A}_{k}\right)^{\text {rec }}X(k)⊆X(Ak)rec .

4.5. Sufficiency of the reciprocity obstruction

Although evidence is scarce, the answer to the following question might always be in the affirmative, as far as one knows:
Question 4.1. Let k k kkk be a finite extension of Q p ( t ) Q p ( t ) Q_(p)(t)\mathbf{Q}_{p}(t)Qp(t). Let X X XXX be a smooth, proper, rationally connected variety over k k kkk. If X ( A k ) rec X A k rec  ≠ ∅ X(A_(k))^("rec ")!=O/X\left(\mathbf{A}_{k}\right)^{\text {rec }} \neq \varnothingX(Ak)rec ≠∅, does it follow that X ( k ) X ( k ) ≠ ∅ X(k)!=O/X(k) \neq \varnothingX(k)≠∅ ?
Question 4.1 has a positive answer when X X XXX is a quadric and p 2 p ≠ 2 p!=2p \neq 2p≠2. Indeed, we recall from Section 4.3 that even X ( A k ) X A k ≠ ∅ X(A_(k))!=O/X\left(\mathbf{A}_{k}\right) \neq \varnothingX(Ak)≠∅ then implies X ( k ) X ( k ) ≠ ∅ X(k)!=O/X(k) \neq \varnothingX(k)≠∅ (see [23]). It also has a positive answer when X X XXX is birationally equivalent to a torsor under a torus over k k kkk. This follows from the work of Harari, Scheiderer, Szamuely, Tian [38, THEOREM 5.1], [97, § 0.3.1] (modulo the comparison between the reciprocity obstruction defined here and the reciprocity obstruction considered in these articles; the latter is weaker, but turns out to suffice to detect rational points on torsors under tori). We note that there are examples of torsors under tori over k k kkk whose smooth compactifications X X XXX satisfy X ( A k ) rec = X A k rec  = ∅ X(A_(k))^("rec ")=O/X\left(\mathbf{A}_{k}\right)^{\text {rec }}=\varnothingX(Ak)rec =∅ while X ( A k ) X A k ≠ ∅ X(A_(k))!=O/X\left(\mathbf{A}_{k}\right) \neq \varnothingX(Ak)≠∅ (see [24, RemarQUE 5.10]). Positive answers to Question 4.1 are known in various other cases in which X X XXX is birationally equivalent to a homogeneous space of a connected linear algebraic group over k k kkk. For specific statements, we refer the reader to the articles quoted in Section 4.3. Question 4.1 remains open in general for smooth compactifications of torsors under connected linear alge-
braic groups over k k kkk, for smooth compactifications of homogeneous spaces of S L n S L n SL_(n)\mathrm{SL}_{n}SLn with finite stabilisers, and for conic bundle surfaces over P k 1 P k 1 P_(k)^(1)\mathbf{P}_{k}^{1}Pk1.
Question 4.1 focuses on the existence of rational points rather than on the density of X ( k ) X ( k ) X(k)X(k)X(k) in X ( A k ) rec X A k rec  X(A_(k))^("rec ")X\left(\mathbf{A}_{k}\right)^{\text {rec }}X(Ak)rec  as the latter property is only known for projective space (see [3, тHEoREM 1]) and hence for varieties that are rational as soon as they possess a rational point, such as quadrics. For smooth compactifications of tori, the density of X ( k ) X ( k ) X(k)X(k)X(k) in X ( A k ) rec X A k rec  X(A_(k))^("rec ")X\left(\mathbf{A}_{k}\right)^{\text {rec }}X(Ak)rec  is known to hold off the set of discrete valuations of k k kkk whose residue field has characteristic p p ppp (see [37, THEOREM 5.2]; for the meaning of "off" here, see [103, DEFINITION 2.9]).
To obtain more positive answers to Question 4.1, it is natural to wish for flexible tools such as general descent theorems and fibration theorems. In the same way that introducing the tight approximation property and replacing Question 3.3 (2) with Question 3.8 was a key step to obtain a problem that behaves well with respect to fibrations into rationally connected varieties (see the discussion in Section 3.2), it is likely that in order to obtain compatibility with descent and fibrations, one will have to strengthen Question 4.1 by incorporating into it a p p ppp-adic analogue of the approximation condition in the Euclidean topology that appears in Definition 3.4. The main challenge, here, is to provide the correct formulation for such a p p ppp-adic tight approximation property.
We note that in any case, a general fibration theorem has to lie deep, as it would presumably give a direct route to the local-global principle for the existence of rational points on smooth projective quadrics over k k kkk (so far unknown when p = 2 p = 2 p=2p=2p=2 ) and hence to the computation of the u u uuu-invariant of k k kkk (equal to 8 ; see Section 4.3). Indeed, in the case of conics over k k kkk, this local-global principle follows from Tate-Lichtenbaum duality [73]; applying a fibration theorem to a general pencil of hyperplane sections of a fixed smooth projective quadric of dimension n 2 n ≥ 2 n >= 2n \geq 2n≥2 would allow one to deduce the general case by induction on n n nnn.

4.6. Further questions

A good understanding of rational points of rationally connected varieties over function fields of curves over p p ppp-adic fields, be it via Question 4.1 or otherwise, should shed light on concrete test questions such as the following:
Questions 4.2. Let p p ppp be a prime number and k k kkk be a finite extension of Q p ( t ) Q p ( t ) Q_(p)(t)\mathbf{Q}_{p}(t)Qp(t).
(1) Does the conjecture of Mináč and Tân on the vanishing of Massey products in Galois cohomology hold for k k kkk ? (See Section 2.5 and [ 78 , 79 ] [ 78 , 79 ] [78,79][78,79][78,79].)
(2) Is there an algorithm that takes as input a smooth, projective, rationally connected variety X X XXX over k k kkk and decides whether X X XXX has a rational point?
One might approach the first of these questions by trying to mimic [49] over k k kkk, which would require making progress on the arithmetic, over k k kkk, of homogeneous spaces of S L n S L n SL_(n)\mathrm{SL}_{n}SLn with finite supersolvable geometric stabilisers.
To put the second question in perspective, let us recall what is known about algorithms for deciding the existence of rational points on arbitrary varieties ("Hilbert's tenth problem") over various fields of interest. Over Q Q Q\mathbf{Q}Q or C ( t ) C ( t ) C(t)\mathbf{C}(t)C(t), the existence of such an algorithm
is an outstanding open problem. Denef [29] showed that over R ( t ) R ( t ) R(t)\mathbf{R}(t)R(t), such an algorithm does not exist. His method was extended to prove that there is no such algorithm over Q p ( t ) Q p ( t ) Q_(p)(t)\mathbf{Q}_{p}(t)Qp(t) (Kim and Roush [62], completed by Degroote and Demeyer [27]), over any finite extension of R ( t ) R ( t ) R(t)\mathbf{R}(t)R(t) that possesses a real place (Moret-Bailly [81]), or, when p 2 p ≠ 2 p!=2p \neq 2p≠2, over any finite extension of Q p ( t ) Q p ( t ) Q_(p)(t)\mathbf{Q}_{p}(t)Qp(t) (Eisenträger [32], Moret-Bailly [81]). In addition, over number fields, it is known that restricting from arbitrary varieties to smooth projective varieties makes no difference (see [96, § II.7], [90, THEOREM 1.1 (I)]). Restricting to smooth, projective, rationally connected varieties, however, does make a drastic difference: Question 4.2 (2) might well have an affirmative answer for all of the fields just mentioned. Over C ( t ) C ( t ) C(t)\mathbf{C}(t)C(t), this is trivially so, by the Graber-Harris-Starr theorem. Over R ( t ) R ( t ) R(t)\mathbf{R}(t)R(t), a positive answer to Question 4.2 (2) would follow from a positive answer to Question 3.8. Indeed, in the notation of Definition 3.4, if X X XXX satisfies the tight approximation property, then X X XXX has a rational point if and only if f | X ( R ) f X ( R ) f|_(X(R))\left.f\right|_{\mathscr{X}(\mathbf{R})}f|X(R) admits a C C ∞ C^(oo)\mathscr{C}^{\infty}C∞ section, a property that can be decided algorithmically. Over number fields, as was observed by Poonen [89, Remark 5.3], a positive answer to Question 4.2 (2) would follow from the conjecture that rational points are always dense in the Brauer-Manin set. It seems likely that a positive answer to Question 4.1 would similarly imply a positive answer to Question 4.2 (2). To mimic Poonen's argument, one runs into the difficulty that the elements of H n r 3 ( X / k , Q / Z ( 2 ) ) H n r 3 ( X / k , Q / Z ( 2 ) ) H_(nr)^(3)(X//k,Q//Z(2))H_{\mathrm{nr}}^{3}(X / k, \mathbf{Q} / \mathbf{Z}(2))Hnr3(X/k,Q/Z(2)) are harder to describe than those of H n r 2 ( X / k , Q / Z ( 1 ) ) = Br ( X ) H n r 2 ( X / k , Q / Z ( 1 ) ) = Br ⁡ ( X ) H_(nr)^(2)(X//k,Q//Z(1))=Br(X)H_{\mathrm{nr}}^{2}(X / k, \mathbf{Q} / \mathbf{Z}(1))=\operatorname{Br}(X)Hnr2(X/k,Q/Z(1))=Br⁡(X), whose interpretation in terms of Azumaya algebras is a key point in [89, REMARK 5.3]; however, this can be remedied by viewing H n r 3 ( X / k , Q / Z ( 2 ) ) H n r 3 ( X / k , Q / Z ( 2 ) ) H_(nr)^(3)(X//k,Q//Z(2))H_{\mathrm{nr}}^{3}(X / k, \mathbf{Q} / \mathbf{Z}(2))Hnr3(X/k,Q/Z(2)), using Bloch-Ogus theory, as the group of
global sections of the Zariski sheaf associated with the presheaf U H e t 3 ( U , Q / Z ( 2 ) ) U ↦ H e t 3 ( U , Q / Z ( 2 ) ) U|->H_(et)^(3)(U,Q//Z(2))U \mapsto H_{\mathrm{et}}^{3}(U, \mathbf{Q} / \mathbf{Z}(2))U↦Het3(U,Q/Z(2)), and describing H e t 3 ( U , Q / Z ( 2 ) ) H e t 3 ( U , Q / Z ( 2 ) ) H_(et)^(3)(U,Q//Z(2))H_{\mathrm{et}}^{3}(U, \mathbf{Q} / \mathbf{Z}(2))Het3(U,Q/Z(2)) via Čech cohomology.

4.7. Other fields

There are a number of other fields over which a better understanding of rational points of rationally connected varieties would be valuable. One of the simplest example is the fraction field k = C ( ( x , y ) ) k = C ( ( x , y ) ) k=C((x,y))k=\mathbf{C}((x, y))k=C((x,y)) of the ring of formal power series C [ [ x , y ] ] C [ [ x , y ] ] C[[x,y]]\mathbf{C}[[x, y]]C[[x,y]], which can be seen as a first step before considering function fields of complex surfaces. This field presents both local and global features, and a reciprocity obstruction can again be defined (in terms of the unramified Brauer group-recall that k k kkk has cohomological dimension 2). This obstruction was used in [24] to produce the first example of a torsor Y Y YYY under a torus, over k k kkk, such that Y ( k ) = Y ( k ) = ∅ Y(k)=O/Y(k)=\varnothingY(k)=∅ but Y ( k v ) Y k v ≠ ∅ Y(k_(v))!=O/Y\left(k_{v}\right) \neq \varnothingY(kv)≠∅ for every discrete valuation v v vvv on k k kkk. The analogues of Question 4.1 and of Questions 4.2 can be asked over this field, too. It is not known, however, whether the reciprocity obstruction explains the absence of rational points on smooth proper varieties that are birationally equivalent to torsors under tori over k k kkk (though see [59, CORoLLAIRE 4.4] for a closely related result involving possibly ramified Brauer classes). We refer the interested reader to [ 18 , 22 , 59 , 60 ] [ 18 , 22 , 59 , 60 ] [18,22,59,60][18,22,59,60][18,22,59,60] for the state of the art.

ACKNOWLEDGMENTS

I am grateful to Jean-Louis Colliot-Thélène, Antoine Ducros, Hélène Esnault, János Kollár, and Ján Mináč for their comments on a first version of the text, and to Olivier
Benoist and Yonatan Harpaz for the pleasant collaborations that have led to the results reported on in this article.

REFERENCES

[1] D. Abramovich, J. Denef, and K. Karu, Weak toroidalization over non-closed fields. Manuscripta math. 142 (2013), no. 1-2, 257-271.
[2] C. Araujo and J. Kollár, Rational curves on varieties. In Higher dimensional varieties and rational points (Budapest, 2001), pp. 13-68, Bolyai Soc. Math. Stud. 12, Springer, Berlin, 2003.
[3] E. Artin and G. Whaples, Axiomatic characterization of fields by the product formula for valuations. Bull. Amer. Math. Soc. 51 (1945), 469-492.
[4] O. Benoist and O. Wittenberg, On the integral Hodge conjecture for real varieties, I. Invent. math. 222 (2020), no. 1, 1-77.
[5] O. Benoist and O. Wittenberg, On the integral Hodge conjecture for real varieties, II. J. Éc. Polytech. Math. 7 (2020), 373-429.
[6] O. Benoist and O. Wittenberg, The tight approximation property. J. reine angew. Math. 776 (2021), 151-200.
[7] S. Bloch and A. Ogus, Gersten's conjecture and the homology of schemes. Ann. Sci. Éc. Norm. Supér. (4) 7 (1974), 181-201.
[8] J. Bochnak and W. Kucharz, The Weierstrass approximation theorem for maps between real algebraic varieties. Math. Ann. 314 (1999), no. 4, 601-612.
[9] A. Borel and A. Haefliger, La classe d'homologie fondamentale d'un espace analytique. Bull. Soc. Math. France 89 (1961), 461-513.
[10] M. Borovoi, The Brauer-Manin obstructions for homogeneous spaces with connected or abelian stabilizer. J. reine angew. Math. 473 (1996), 181-194.
[11] S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron models. Ergeb. Math. Grenzgeb. (3) 21, Springer, Berlin, 1990.
[12] F. Campana, Connexité rationnelle des variétés de Fano. Ann. Sci. Éc. Norm. Supér. (4) 25 (1992), no. 5, 539-545.
[13] Y. Cao, Approximation forte pour les variétés avec une action d'un groupe linéaire. Compos. Math. 154 (2018), no. 4, 773-819.
[14] J.-L. Colliot-Thélène, Groupes linéaires sur les corps de fonctions de courbes réelles. J. reine angew. Math. 474 (1996), 139-167.
[15] J.-L. Colliot-Thélène, Rational connectedness and Galois covers of the projective line. Ann. of Math. (2) 151 (2000), no. 1, 359-373.
[16] J.-L. Colliot-Thélène, Variétés presque rationnelles, leurs points rationnels et leurs dégénérescences. In Arithmetic geometry, pp. 1-44, Lecture Notes in Math. 2009, Springer, Berlin, 2011.
[17] J.-L. Colliot-Thélène and P. Gille, Remarques sur l'approximation faible sur un corps de fonctions d'une variable. In Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), pp. 121-134, Progr. Math. 226, Birkhäuser Boston, Boston, MA, 2004.
[18] J.-L. Colliot-Thélène, P. Gille, and R. Parimala, Arithmetic of linear algebraic groups over 2-dimensional geometric fields. Duke Math. J. 121 (2004), no. 2, 285-341.
[19] J.-L. Colliot-Thélène, D. Harbater, J. Hartmann, D. Krashen, R. Parimala, and V. Suresh, Local-global principles for zero-cycles on homogeneous spaces over arithmetic function fields. Trans. Amer. Math. Soc. 372 (2019), no. 8, 5263-5286.
[20] J.-L. Colliot-Thélène, D. Harbater, J. Hartmann, D. Krashen, R. Parimala, and V. Suresh, Local-global principles for tori over arithmetic curves. Algebr. Geom. 7 (2020), no. 5, 607-633.
[21] J.-L. Colliot-Thélène, D. Harbater, J. Hartmann, D. Krashen, R. Parimala, and V. Suresh, Local-global principles for constant reductive groups over semi-global fields. 2021, arXiv:2108.12349.
[22] J.-L. Colliot-Thélène, M. Ojanguren, and R. Parimala, Quadratic forms over fraction fields of two-dimensional Henselian rings and Brauer groups of related schemes. In Algebra, arithmetic and geometry, Part I, II (Mumbai, 2000), pp. 185-217, Tata Inst. Fund. Res. Stud. Math. 16, Tata Inst. Fund. Res, Bombay, 2002.
[23] J.-L. Colliot-Thélène, R. Parimala, and V. Suresh, Patching and local-global principles for homogeneous spaces over function fields of p p ppp-adic curves. Comment. Math. Helv. 87 (2012), no. 4, 1011-1033.
[24] J.-L. Colliot-Thélène, R. Parimala, and V. Suresh, Lois de réciprocité supérieures et points rationnels. Trans. Amer. Math. Soc. 368 (2016), no. 6, 4219-4255.
[25] J.-L. Colliot-Thélène and J.-J. Sansuc, La descente sur les variétés rationnelles. II. Duke Math. J. 54 (1987), no. 2, 375-492.
[26] J.-L. Colliot-Thélène and J.-J. Sansuc, The rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group). In Algebraic groups and homogeneous spaces, pp. 113-186, Tata Inst. Fund. Res. Stud. Math. 19, Tata Inst. Fund. Res, Mumbai, 2007.
[27] C. Degroote and J. Demeyer, Hilbert's tenth problem for rational function fields over p p ppp-adic fields. J. Algebra 361 (2012), 172-187.
[28] C. Demarche, G. Lucchini Arteche, and D. Neftin, The Grunwald problem and approximation properties for homogeneous spaces. Ann. Inst. Fourier (Grenoble) 67 (2017), no. 3, 1009-1033.
[29] J. Denef, The Diophantine problem for polynomial rings and fields of rational functions. Trans. Amer. Math. Soc. 242 (1978), 391-399.
[30] A. Ducros, Fibrations en variétés de Severi-Brauer au-dessus de la droite projective sur le corps des fonctions d'une courbe réelle. C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), no. 1, 71-75.
[31] A. Ducros, L'obstruction de réciprocité à l'existence de points rationnels pour certaines variétés sur le corps des fonctions d'une courbe réelle. J. reine angew. Math. 504 (1998), 73-114.
[32] K. Eisenträger, Hilbert's tenth problem for function fields of varieties over number fields and p p ppp-adic fields. J. Algebra 310 (2007), no. 2, 775-792.
[33] T. Ekedahl, An effective version of Hilbert's irreducibility theorem. In Séminaire de Théorie des Nombres, Paris 1988-1989, pp. 241-249, Progr. Math. 91, Birkhäuser Boston, Boston, MA, 1990.
[34] T. Graber, J. Harris, and J. Starr, Families of rationally connected varieties. J. Amer. Math. Soc. 16 (2003), no. 1, 57-67.
[35] D. Harari, Méthode des fibrations et obstruction de Manin. Duke Math. J. 75 (1994), no. 1, 221-260.
[36] D. Harari, Quelques propriétés d'approximation reliées à la cohomologie galoisienne d'un groupe algébrique fini. Bull. Soc. Math. France 135 (2007), no. 4, 549 564 549 − 564 549-564549-564549−564.
[37] D. Harari, C. Scheiderer, and T. Szamuely, Weak approximation for tori over p-adic function fields. Int. Math. Res. Not. IMRN 10 (2015), 2751-2783.
[38] D. Harari and T. Szamuely, Local-global questions for tori over p p ppp-adic function fields. J. Algebraic Geom. 25 (2016), no. 3, 571-605.
[39] D. Harbater, Galois coverings of the arithmetic line. In Number theory, pp. 165-195, Lecture Notes in Math. 1240, Springer, Berlin, 1987.
[40] D. Harbater and J. Hartmann, Patching over fields. Israel J. Math. 176 (2010), 61-107.
[41] D. Harbater, J. Hartmann, V. Karemaker, and F. Pop, A comparison between obstructions to local-global principles over semiglobal fields. 2019, arXiv:1903.08007.
[42] D. Harbater, J. Hartmann, and D. Krashen, Applications of patching to quadratic forms and central simple algebras. Invent. math. 178 (2009), no. 2, 231-263.
[43] D. Harbater, J. Hartmann, and D. Krashen, Local-global principles for Galois cohomology. Comment. Math. Helv. 89 (2014), no. 1, 215-253.
[44] D. Harbater, J. Hartmann, and D. Krashen, Local-global principles for torsors over arithmetic curves. Amer. J. Math. 137 (2015), no. 6, 1559-1612.
[45] D. Harbater, J. Hartmann, D. Krashen, R. Parimala, and V. Suresh, Local-global Galois theory of arithmetic function fields. Israel J. Math. 232 (2019), no. 2, 849-882.
[46] D. Harbater, D. Krashen, and A. Pirutka, Local-global principles for curves over semi-global fields. Bull. Lond. Math. Soc. 53 (2021), no. 1, 177-193.
[47] Y. Harpaz, D. Wei, and O. Wittenberg, Rational points on fibrations with few nonsplit fibres. 2021, arXiv:2109.03547.
[48] Y. Harpaz and O. Wittenberg, On the fibration method for zero-cycles and rational points. Ann. of Math. (2) 183 (2016), no. 1, 229-295.
[49] Y. Harpaz and O. Wittenberg, The Massey vanishing conjecture for number fields. 2019, arXiv:1904.06512, to appear, Duke Math. J.
[50] Y. Harpaz and O. Wittenberg, Zéro-cycles sur les espaces homogènes et problème de Galois inverse. J. Amer. Math. Soc. 33 (2020), no. 3, 775-805.
[51] Y. Harpaz and O. Wittenberg, Supersolvable descent for rational points. Preprint, 2021.
[52] B. Hassett, Weak approximation and rationally connected varieties over function fields of curves. In Variétés rationnellement connexes: aspects géométriques et arithmétiques, pp. 115-153, Panor. Synthèses 31, Soc. Math. France, Paris, 2010.
[53] B. Hassett and Y. Tschinkel, Weak approximation over function fields. Invent. math. 163 (2006), no. 1, 171-190.
[54] D. R. Heath-Brown, Zeros of systems of p p p\mathfrak{p}p-adic quadratic forms. Compos. Math. 146 (2010), no. 2, 271-287.
[55] A. Hogadi and C. Xu, Degenerations of rationally connected varieties. Trans. Amer. Math. Soc. 361 (2009), no. 7, 3931-3949.
[56] Y. Hu, Weak approximation over function fields of curves over large or finite fields. Math. Ann. 348 (2010), no. 2, 357-377.
[57] Y. Hu, Hasse principle for simply connected groups over function fields of surfaces. J. Ramanujan Math. Soc. 29 (2014), no. 2, 155-199.
[58] Y. Hu, A cohomological Hasse principle over two-dimensional local rings. Int. Math. Res. Not. IMRN 14 (2017), 4369-4397.
[59] D. Izquierdo, Dualité et principe local-global pour les anneaux locaux henséliens de dimension 2. Algebr. Geom. 6 (2019), no. 2, 148-176.
[60] D. Izquierdo and G. Lucchini Arteche, Local-global principles for homogeneous spaces over some two-dimensional geometric global fields. 2021, arXiv:2101.08245, to appear, J. reine angew. Math.
[61] K. Kato, A Hasse principle for two-dimensional global fields. J. reine angew. Math. 366 (1986), 142-183.
[62] K. H. Kim and F. W. Roush, Diophantine unsolvability over p p ppp-adic function fields. J. Algebra 176 (1995), no. 1, 83-110.
[63] J. Kollár, Rational curves on algebraic varieties. Ergeb. Math. Grenzgeb. (3) 32, Springer, Berlin, 1996.
[64] J. Kollár, Rationally connected varieties over local fields. Ann. of Math. (2) 150 (1999), no. 1, 357-367.
[65] J. Kollár, Rationally connected varieties and fundamental groups. In Higher dimensional varieties and rational points (Budapest, 2001), pp. 69-92, Bolyai Soc. Math. Stud. 12, Springer, Berlin, 2003.
[66] J. Kollár, Specialization of zero cycles. Publ. Res. Inst. Math. Sci. 40 (2004), no. 3, 689-708.
[67] J. Kollár, Holomorphic and pseudo-holomorphic curves on rationally connected varieties. Port. Math. 67 (2010), no. 2, 155-179.
[68] J. Kollár and F. Mangolte, Approximating curves on real rational surfaces. J. Algebraic Geom. 25 (2016), no. 3, 549-570.
[69] J. Kollár, Y. Miyaoka, and S. Mori, Rationally connected varieties. J. Algebraic Geom. 1 (1992), no. 3, 429-448.
[70] S. Lang, The theory of real places. Ann. of Math. (2) 57 (1953), no. 2, 378-391.
[71] S. Lang, Higher dimensional diophantine problems. Bull. Amer. Math. Soc. 80 (1974), 779-787.
[72] D. B. Leep, The u u uuu-invariant of p p ppp-adic function fields. J. reine angew. Math. 679 (2013), 65-73.
[73] S. Lichtenbaum, Duality theorems for curves over p p ppp-adic fields. Invent. math. 7 (1969), 120-136.
[74] Q. Liu, Tout groupe fini est un groupe de Galois sur Q p ( T ) Q p ( T ) Q_(p)(T)\mathbf{Q}_{p}(T)Qp(T), d'après Harbater. In Recent developments in the inverse Galois problem, pp. 261-265, Contemp. Math. 186, Amer. Math. Soc., Providence, RI, 1995.
[75] G. Lucchini Arteche, The unramified Brauer group of homogeneous spaces with finite stabilizer. Trans. Amer. Math. Soc. 372 (2019), no. 8, 5393-5408.
[76] Yu. I. Manin, Le groupe de Brauer-Grothendieck en géométrie diophantienne. In Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, pp. 401-411, Gauthier-Villars, Paris, 1970.
[77] V. Mehmeti, Patching over Berkovich curves and quadratic forms. Compos. Math. 155 (2019), no. 12, 2399-2438.
[78] J. Mináč and N. D. Tân, Triple Massey products vanish over all fields. J. Lond. Math. Soc. (2) 94 (2016), no. 3, 909-932.
[79] J. Mináč and N. D. Tân, Triple Massey products and Galois theory. J. Eur. Math. Soc. (JEMS) 19 (2017), no. 1, 255-284.
[80] L. Moret-Bailly, Construction de revêtements de courbes pointées. J. Algebra 240 (2001), no. 2, 505-534
[81] L. Moret-Bailly, Elliptic curves and Hilbert's tenth problem for algebraic function fields over real and p p ppp-adic fields. J. reine angew. Math. 587 (2005), 77-143.
[82] J. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of number fields. 2nd edn. Grundlehren Math. Wiss. 323, Springer, Berlin, 2008.
[83] A. Pál and T. Schlank, The Brauer-Manin obstruction to the local-global principle for the embedding problem. 2016, arXiv:1602.04998.
[84] A. Pál and E. Szabó, The fibration method over real function fields. Math. Ann. 378 (2020), no. 3-4, 993-1019.
[85] R. Parimala, R. Preeti, and V. Suresh, Local-global principle for reduced norms over function fields of p p ppp-adic curves. Compos. Math. 154 (2018), no. 2, 410-458.
[86] R. Parimala and V. Suresh, The u u uuu-invariant of the function fields of p p ppp-adic curves. Ann. of Math. (2) 172 (2010), no. 2, 1391-1405.
[87] R. Parimala and V. Suresh, Period-index and u u uuu-invariant questions for function fields over complete discretely valued fields. Invent. math. 197 (2014), no. 1, 215 235 215 − 235 215-235215-235215−235.
[88] R. Parimala and V. Suresh, Local-global principle for unitary groups over function fields of p p ppp-adic curves. 2020, arXiv:2004.10357.
[89] B. Poonen, Heuristics for the Brauer-Manin obstruction for curves. Exp. Math. 15 (2006), no. 4, 415-420.
[90] B. Poonen, Existence of rational points on smooth projective varieties. J. Eur. Math. Soc. (JEMS) 11 (2009), no. 3, 529-543.
[91] F. Pop, Embedding problems over large fields. Ann. of Math. (2) 144 (1996), no. 1, 1-34.
[92] R. Preeti, Classification theorems for Hermitian forms, the Rost kernel and Hasse principle over fields with c d 2 ( k ) 3 c d 2 ( k ) ≤ 3 cd_(2)(k) <= 3c d_{2}(k) \leq 3cd2(k)≤3. J. Algebra 385 (2013), 294-313.
[93] F. Russo, The antibirational involutions of the plane and the classification of real del Pezzo surfaces. In Algebraic geometry, pp. 289-312, de Gruyter, Berlin, 2002.
[94] C. Scheiderer, Hasse principles and approximation theorems for homogeneous spaces over fields of virtual cohomological dimension one. Invent. math. 125 (1996), no. 2, 307-365.
[95] J.-P. Serre, Topics in Galois theory. 2nd edn. Res. Notes Math. 1, A K Peters, Ltd., Wellesley, MA, 2008.
[96] C. Smoryński, Logical number theory. I. Universitext. Springer, Berlin, 1991.
[97] Y. Tian, Arithmétique des groupes algébriques au-dessus du corps des fonctions d'une courbe sur un corps p-adique. Ph.D. thesis, Université Paris-Sud, 2020. https://tel.archives-ouvertes.fr/tel-02985977.
[98] Y. Tian, Obstructions to weak approximation for reductive groups over p p ppp-adic function fields. J. Number Theory 220 (2021), 128-162.
[99] Z. Tian, Weak approximation for cubic hypersurfaces. Duke Math. J. 164 (2015), no. 7, 1401-1435.
[100] Z. Tian and H. R. Zong, One-cycles on rationally connected varieties. Compos. Math. 150 (2014), no. 3, 396-408.
[101] C. Voisin, Some aspects of the Hodge conjecture. Jpn. J. Math. 2 (2007), no. 2, 261-296.
[102] O. Wittenberg, On Albanese torsors and the elementary obstruction. Math. Ann. 340 (2008), no. 4, 805-838.
[103] O. Wittenberg, Rational points and zero-cycles on rationally connected varieties over number fields. In Algebraic geometry: Salt Lake City 2015, pp. 597-635, Proc. Sympos. Pure Math. 97, Amer. Math. Soc., Providence, RI, 2018.

OLIVIER WITTENBERG

Institut Galilée, Université Sorbonne Paris Nord, 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse, France, wittenberg @ math.univ-paris13.fr

LIST OF CONTRIBUTORS

Abért, Miklós 5:3374
Aganagic, Mina 3:2108
Andreev, Nikolai 1:322
Ardila-Mantilla, Federico 6:4510
Asok, Aravind 3:2146
Bach, Francis 7:5398
Baik, Jinho 6:4190
Ball, Keith 4:3104
Bamler, Richard H. 4:2432
Bansal, Nikhil 7:5178
Bao, Gang 7:5034
Barreto, Andre 6:4800
Barrow-Green, June 7:5748
Bauerschmidt, Roland 5:3986
Bayer, Arend 3:2172
Bedrossian, Jacob 7:5618
Beliaev, Dmitry 1:V
Berger, Marsha J. 7:5056
Berman, Robert J. 4:2456
Bestvina, Mladen 2:678
Beuzart-Plessis, Raphaël 3:1712
Bhatt, Bhargav 2 : 712 2 : 712 2:7122: 7122:712
Binyamini, Gal 3:1440
Blumenthal, Alex 7:5618
Bodineau, Thierry 2:750
Bonetto, Federico 5:4010
Böttcher, Julia 6:4542
Braverman, Alexander 2:796
Braverman, Mark 1:284
Brown, Aaron 5:3388
Buckmaster, Tristan 5:3636
Burachik, Regina S. 7:5212
Burger, Martin 7:5234
Buzzard, Kevin 2:578
Calegari, Danny 4:2484
Calegari, Frank 2:610
Caprace, Pierre-Emmanuel 3:1554
Caraiani, Ana 3:1744
Cardaliaguet, Pierre 5:3660
Carlen, Eric 5:4010
Cartis, Coralia 7:5256
Chaika, Jon 5:3412
Champagnat, Nicolas 7:5656
Chizat, Lénaïc 7:5398
Cieliebak, Kai 4:2504
Cohn, Henry 1:82
Colding, Tobias Holck 2:826
Collins, Benoît 4:3142
Dai, Yu-Hong 7:5290
Darmon, Henri 1:118
Dasgupta, Samit 3:1768
de la Salle, Mikael 4:3166
De Lellis, Camillo 2:872
Delarue, François 5:3660
Delecroix, Vincent 3:2196
Demers, Mark F. 5:3432
Ding, Jian 6:4212
Dobrinen, Natasha 3:1462
Dong, Bin 7:5420
Drivas, Theodore D. 5:3636
Du, Xiumin 4:3190
Dubédat, Julien 6:4212
Dujardin, Romain 5:3460
Duminil-Copin, Hugo 1:164
Dwork, Cynthia 6:4740
Dyatlov, Semyon 5:3704
E, Weinan 2:914
Efimov, Alexander I. 3:2212
Eldan, Ronen 6:4246
Etheridge, Alison 6:4272
Fasel, Jean 3:2146
Feigin, Evgeny 4:2930
Ferreira, Rita 5:3724
Fisher, David 5:3484
Fonseca, Irene 5:3724
Fournais, Søren 5:4026
Frank, Rupert L. 1:142, 5:3756
Friedgut, Ehud 6:4568
Funaki, Tadahisa 6:4302
Gallagher, Isabelle 2:750
Gamburd, Alexander 3:1800
Gentry, Craig 2:956
Georgieva, Penka 4:2530
Giuliani, Alessandro 5:4040
Gonçalves, Patrícia 6:4326
Gotlib, Roy 6:4842
Goujard, Élise 3:2196
Gould, Nicholas I. M. 7:5256
Grima, Clara I. 7:5702
Guionnet, Alice 2:1008
Gupta, Neena 3:1578
Guth, Larry 2:1054
Gwynne, Ewain 6:4212
Habegger, Philipp 3:1838
Hairer, Martin 1:26
Hastings, Matthew B. 5:4074
Hausel, Tamás 3:2228
Helmuth, Tyler 5:3986
Hesthaven, Jan S. 7:5072
Higham, Nicholas J. 7:5098
Hintz, Peter 5:3924
Holden, Helge 1:11
Holzegel, Gustav 5:3924
Hom, Jennifer 4:2740
Houdayer, Cyril 4:3202
Huh, June 1:212
Ichino, Atsushi 3:1870
Imhausen, Annette 7:5772
Ionescu, Alexandru D. 5:3776
Iritani, Hiroshi 4:2552
Isaksen, Daniel C. 4:2768
Jackson, Allyn 1:548, 1:554

1 : 560 , 1 : 566 1 : 560 , 1 : 566 1:560,1:5661: 560,1: 5661:560,1:566

Jain, Aayush 6:4762
Jegelka, Stefanie 7:5450
Jia, Hao 5:3776
Jitomirskaya, Svetlana 2:1090
Kakde, Mahesh 3:1768
Kalai, Gil 1:50
Kaletha, Tasho 4:2948
Kamnitzer, Joel 4:2976
Kang, Hyeonbae 7:5680
Kato, Syu 3:1600
Kaufman, Tali 6:4842
Kazhdan, David 2:796
Kenig, Carlos 1:5, 1:9
Kleiner, Bruce 4:2376
Klingler, Bruno 3:2250
Knutson, Allen 6:4582
Koukoulopoulos, Dimitris 3:1894
Kozlowski, Karol Kajetan 5:4096
Krichever, Igor 2:1122
Kutyniok, Gitta 7:5118
Kuznetsov, Alexander 2:1154
Lacoin, Hubert 6:4350
Larsen, Michael J. 3:1624
Lemańczyk, Mariusz 5:3508
Lepski, Oleg V. 7:5478
LeVeque, Randall J. 7:5056
Levine, Marc 3:2048
Lewin, Mathieu 5:3800
Li, Chi 3:2286
Lin, Huijia 6:4762
Liu, Gang 4:2576
Liu, Yi 4:2792
Loeffler, David 3:1918
Loss, Michael 5:4010
Lü, Qi 7:5314
Lugosi, Gábor 7:5500
Luk, Jonathan 5:4120
Macrì, Emanuele 3:2172
Mann, Kathryn 4:2594
Marks, Andrew S. 3:1488
Maynard, James 1:240
McLean, Mark 4:2616
Méléard, Sylvie 7:5656
Mikhailov, Roman 4:2806
Mohammadi, Amir 5:3530
Mossel, Elchanan 6:4170
Nakanishi, Kenji 5:3822
Nazarov, Alexander I. 5:3842
Neeman, Amnon 3:1636
Nelson, Jelani 6:4872
Nickl, Richard 7:5516
Nikolaus, Thomas 4:2826
Norin, Sergey 6:4606
Novik, Isabella 6:4622
Novikov, Dmitry 3:1440
Ogata, Yoshiko 5:4142
Okounkov, Andrei 1:376, 1:414
1 : 460 , 1 : 492 1 : 460 , 1 : 492 1:460,1:4921: 460,1: 4921:460,1:492
Ozdaglar, Asuman 7:5340
Pagliantini, Cecilia 7:5072
Panchenko, Dmitry 6:4376
Paternain, Gabriel P. 7:5516
Peeva, Irena 3:1660
Perelman, Galina 5:3854
Pierce, Lillian B. 3:1940
Pixton, Aaron 3:2312
Pramanik, Malabika 4:3224
Pretorius, Frans 2:652
Procesi, Michela 5:3552
Prokhorov, Yuri 3:2324
Punshon-Smith, Sam 7:5618
Ramanan, Kavita 6:4394
Ramasubramanian, Krishnamurthi 7:5784
Randal-Williams, Oscar 4:2856
Rasmussen, Jacob 4:2880
Raz, Ran 1:106
Regev, Oded 6:4898
Remenik, Daniel 6:4426
Ripamonti, Nicolò 7:5072
Safra, Muli (Shmuel) 6:4914
Sahai, Amit 6:4762
Saint-Raymond, Laure 2:750
Sakellaridis, Yiannis 4:2998
Saloff-Coste, Laurent 6:4452
Sayin, Muhammed O. 7:5340
Schacht, Mathias 6:4646
Schechtman, Gideon 4:3250
Schölkopf, Bernhard 7:5540
Schwartz, Richard Evan 4:2392
Scott, Alex 6:4660
Sfard, Anna 7:5716
Shan, Peng 4:3038
Shapira, Asaf 6:4682
Sheffield, Scott 2:1202
Shin, Sug Woo 3:1966
Shkoller, Steve 5:3636
Shmerkin, Pablo 4:3266
Silver, David 6:4800
Silverman, Joseph H. 3:1682
Simonella, Sergio 2:750
Smirnov, Stanislav 1:V
Solovej, Jan Philip 5:4026
Soundararajan, Kannan 1:66, 2:1260
Stroppel, Catharina 2:1312
Sturmfels, Bernd 6:4820
Sun, Binyong 4:3062
Svensson, Ola 6:4970
Taimanov, Iskander A. 4:2638
Tarantello, Gabriella 5:3880
Tian, Ye 3:1990
Tikhomirov, Konstantin 4:3292
Toint, Philippe L. 7:5296
Tokieda, Tadashi 1:160
Tran, Viet Chi 7:5656
Tucsnak, Marius 7:5374
Ulcigrai, Corinna 5:3576
Van den Bergh, Michel 2:1354
Varjú, Péter P. 5:3610
Venkatraman, Raghavendra 5:3724
Viazovska, Maryna 1:270
Vicol, Vlad 5:3636
Vidick, Thomas 6:4996
Vignéras, Marie-France 1:332
von Kügelgen, Julius 7:5540
Wahl, Nathalie 4:2904
Wang, Guozhen 4:2768
Wang, Lu 4:2656
Wang, Weiqiang 4:3080
Ward, Rachel 7:5140
Wei, Dongyi 5:3902
Weiss, Barak 5:3412
White, Stuart 4:3314
Wigderson, Avi 2:1392
Williams, Lauren K. 6:4710
Willis, George A. 3:1554
Wittenberg, Olivier 3:2346
Wood, Melanie Matchett 6:4476
Xu, Zhouli 4:2768
Ying, Lexing 7:5154
Yokoyama, Keita 3:1504
Young, Robert J. 4:2678
Zerbes, Sarah Livia 3:1918
Zhang, Cun-Hui 7:5594
Zhang, Kaiqing 7:5340
Zhang, Zhifei 5:3902
Zheng, Tianyi 4:3340
Zhou, Xin 4:2696
Zhu, Chen-Bo 4:3062
Zhu, Xiaohua 4:2718
Zhu, Xinwen 3:2012
Zhuk, Dmitriy 3:1530
Zograf, Peter 3:2196
Zorich, Anton 3:2196
P RES S

  1. 2 We encode T T TTT, e.g., by its recursive index.
    3 In [32], Paris showed that I t P H 2 3 I t P H 2 3 ItPH_(2)^(3)\mathrm{ItPH}_{2}^{3}ItPH23 is independent of P A P A PA\mathrm{PA}PA, while his argument implies the equivalence of statements 2 and 3. See Section 5.2.
  2. 5 See Section 5 for the construction of the moduli space M d N M d N M_(d)^(N)\mathcal{M}_{d}^{N}MdN.
  3. 6 Although in fairness it should be noted that [91] suggests the opposite conclusion, stating: "Are there any rational periodic orbits of a quadratic x 2 + c x 2 + c x^(2)+cx^{2}+cx2+c of period greater than 3? The results for periods 1,2 , and 3 would lead one to suspect that there must be."
  4. 7 We remark that it is easy to prove uniform boundedness for x d + c x d + c x^(d)+cx^{d}+cxd+c over Q Q Q\mathbb{Q}Q when d d ddd is odd, and more generally over any field K / Q K / Q K//QK / \mathbb{Q}K/Q with a real embedding. Indeed, it is an elementary fact that if f : R R f : R → R f:RrarrRf: \mathbb{R} \rightarrow \mathbb{R}f:R→R is any nondecreasing function, then f f fff has no nonfixed periodic points; cf. [64].
    8 The gonality of an algebraic curve X X XXX, or its function field, is the minimal degree of a nonconstant map X P 1 X → P 1 X rarrP^(1)X \rightarrow \mathbb{P}^{1}X→P1.
  5. 1 Of course, the following discussion also applies to U ( W ) U ( W ) U(W)U(W)U(W).
    2 A Whittaker datum of U ( V ) U V ′ U(V^('))U\left(V^{\prime}\right)U(V′) is a pair ( N , θ ) ( N , θ ) (N,theta)(N, \theta)(N,θ) consisting of a maximal unipotent subgroup
    N U ( V ) N ⊂ U V ′ N sub U(V^('))N \subset U\left(V^{\prime}\right)N⊂U(V′) and a generic character θ : N C × Î¸ : N → C × theta:N rarrC^(xx)\theta: N \rightarrow \mathbb{C}^{\times}θ:N→C×. This datum only matters up to conjugacy.
  6. 1 We make a small abuse of notation by using X X XXX to denote both the conjugacy class from the introduction, which is used in the definition of a Shimura datum, and the quotient G ( R ) / K A G ( R ) / K ∞ ∘ A ∞ ∘ G(R)//K_(oo)^(@)A_(oo)^(@)G(\mathbb{R}) / K_{\infty}^{\circ} A_{\infty}^{\circ}G(R)/K∞∘A∞∘ considered in this section. See [ 2 4 $ § 2 . 4 ] [ 2 4 $ § 2 . 4 ] [24$§2.4][\mathbf{2 4} \mathbf{\$} \mathbf{\S} \mathbf{2 . 4}][24$§2.4] for an extended discussion of the various quotients.
  7. 3 As a consequence of the comparison with moduli spaces of local shtukas in [56], one obtains a group-theoretic characterization of Rapoport-Zink spaces as local Shimura varieties determined by the tuple ( G , b , μ ) ( G , b , μ ) (G,b,mu)(G, b, \mu)(G,b,μ). We suppress ( G , μ ) ( G , μ ) (G,mu)(G, \mu)(G,μ) from the notation for simplicity.
  8. 4 The result precedes the notion of diamonds and, in order to ensure that S K p b S K p ∘ b S_(K^(p))^(@b)S_{K^{p}}^{\circ b}SKp∘b is a diamond, one needs to take care in defining it. At hyperspecial level, one should consider the adic generic fiber of the formal completion of the integral model of the Shimura variety along the Newton stratum indexed by b b bbb in its special fiber.
  9. 2 See [125] and [81] for definition and properties of expanders.
    3 Let p p ppp be a large prime and denote by X ( p ) = X ( p ) ( 0 , 0 , 0 ) X ∗ ( p ) = X ( p ) ∖ ( 0 , 0 , 0 ) X^(**)(p)=X(p)\\(0,0,0)X^{*}(p)=X(p) \backslash(0,0,0)X∗(p)=X(p)∖(0,0,0) the solutions of (1.1) modulo p p ppp with the removal of ( 0 , 0 , 0 ) ( 0 , 0 , 0 ) (0,0,0)(0,0,0)(0,0,0). The Markoff graphs are obtained by joining each x x xxx in X ( p ) X ∗ ( p ) X^(**)(p)X^{*}(p)X∗(p) to R j ( x ) , j = 1 , 2 , 3 R j ( x ) , j = 1 , 2 , 3 R_(j)(x),j=1,2,3R_{j}(x), j=1,2,3Rj(x),j=1,2,3. They were considered first by Arthur Baragar in his thesis [3].
  10. 6 The exponent 1 3 1 3 (1)/(3)\frac{1}{3}13 in (1.5) has been improved to 7 9 7 9 (7)/(9)\frac{7}{9}79 in [87].
    7 We remark that in [52] Corvaja and Zannier showed that the greatest prime factor of x y x y xyx yxy for a Markoff triple ( x , y , z ) ( x , y , z ) (x,y,z)(x, y, z)(x,y,z) tends to infinity.
  11. 8 "Trotz der außerordentlich merkwürdigen und wichtigen Resultate scheinen diese schwiergen Untersuchungen wenig bekannt zu sein" [In spite of the extraordinarily noteworthy and important results these difficult investigations seem to be little known]
  12. 2 More precisely, we obtain a relation to the first derivative L ( V π , 1 ) L ′ V Ï€ ∗ , 1 L^(')(V_(pi)^(**),1)L^{\prime}\left(V_{\pi}^{*}, 1\right)L′(Vπ∗,1), with V π V Ï€ V_(pi)V_{\pi}VÏ€ being 1critical. Unfortunately, this computation does not give us any information about the étale class in H 1 ( E , V π ) H 1 E , V Ï€ H^(1)(E,V_(pi))H^{1}\left(E, V_{\pi}\right)H1(E,VÏ€), since the motivic class might be in the kernel of the étale realisation map. (This is the fundamental obstruction to proving the Bloch-Kato conjecture for 1critical Galois representations.)
  13. 3 This argument can be used to construct an Eichler-Shimura isomorphism in families for G S p 4 G S p 4 GSp_(4)\mathrm{GSp}_{4}GSp4, which interpolates the classical H 1 H 1 H^(1)H^{1}H1 comparison isomorphism at almost all classical points - see [51].
  14. 3 Sometimes these methods extend beyond the stated hypotheses. For example, [18] also works for Kisin-Pappas models (Section 6), and [26] proves a uniformization result also in the nonbasic case. However, we will not try to present the methods in their maximally general settings.
  15. 2 If char F = F = F=F=F= char k k kkk (the equal characteristic case), this assumption on k k kkk is not necessary. We impose it here to have a uniform treatment of both equal and mixed characteristic (i.e., char F char k ) F ≠ char ⁡ k ) F!=char k)F \neq \operatorname{char} k)F≠char⁡k) cases. For the same reason, we work with perfect algebraic geometry below even in equal characteristic.
  16. 5 Such restriction functor defines the so-called fusion product, a key concept in the geometric Satake equivalence. The terminology "fusion" originally comes from conformal field theory.
  17. 1 Voevodsky showed this is not possible integrally, so the best one can hope for is a t t ttt structure with Q Q Q\mathbb{Q}Q-coefficients.
  18. 2 In fact, at the beginning of § 3 § 3 §3\S 3§3 of [28], Bloch and Kato write, "The cohomological symbol defined by Tate [114] gives a map . . . which one conjectures to be an isomorphism quite generally."
  19. 1 For example, the square-tiled surface in Figure 1 is made up of 54 squares, has 3 conical points of angle 3 π 3 Ï€ 3pi3 \pi3Ï€ (corresponding to simple zeros of q q qqq ), and 7 conical points of angle π Ï€ pi\piÏ€ (corresponding to simple poles of q q qqq ). Therefore, it has genus 0 and belongs to the principal stratum Q ( 1 3 , 1 7 ) Q 1 3 , − 1 7 Q(1^(3),-1^(7))\mathcal{Q}\left(1^{3},-1^{7}\right)Q(13,−17).
  20. 1 In the classical formulation, Tian's CM weight or, equivalently, the Donaldson-Futaki invariant is used to define the K-stability. However, to fit our discussion in the nonArchimedean framework, we use the equivalent formulation via the M N A M N A M^(NA)\mathbf{M}^{\mathrm{NA}}MNA functional.